Portfolio-performance assessment

ABSTRACT

A portfolio-analysis tool receives data that describe an actual portfolio. It computes from those data the returns or other performance measures of hypothetical portfolios whose holdings are drawn from the assets that the actual portfolio held during some period. Among the purposes of doing so is to detect biases made in investment-portfolio actions of the type taken, for instance, to accommodate cash inflows and withdrawals. For that purpose, differences between the hypothetical portfolio and the actual portfolio are so made as to offset portfolio actions identified by finding differences between the weights that positions actually exhibit and the weights they would result from return only. Returns for the hypothetical portfolio are computed by calculating an offset return incrementally, one such offset at a time, and then computing the hypothetical portfolio&#39;s return as the sum of quantities proportional to the offset return and that of the actual portfolio.

CROSS-REFERENCE TO RELATED APPLICATIONS

The present application is related to two commonly assigned copendingU.S. patent applications that were filed on the same day as this one byHarold J. A. Haig and are hereby incorporated by reference. One isentitled Determining Portfolio Performance Measures by Weight-BasedAction Detection, and the other is entitledHypothetical-Portfolio-Return Determination.

BACKGROUND OF THE INVENTION

1. Field of the Invention

The present invention is directed to investment-portfolio assessment.

2. Background Information

Many analytical tools have been developed to help improve portfolioperformance. Some, which are not the focus of the invention to bedescribed below, deal with trading. Such tools come into play after adecision has been made to buy or sell a given asset. An essential goalof trading is to buy or sell that asset in such a manner that one's ownactions do not affect the traded asset's market price. An effectivetrade, in concept, would occur at a price equal to what the market pricewould have been in the absence of the trade. Algorithmic trading is oneexample of several existing methods used to diffuse trades in such amanner as to minimize or eliminate adverse price impacts.

Distinct from trading, which presupposes decisions to buy and sell, isthe portfolio-management domain, to which the invention below isdirected, where the decisions about which assets to buy and sell aremade. Managing a portfolio of financial assets may involve many theoriesand principles in finance and practices from investment management.Generally, a portfolio manager gathers asset data, current market-trenddata, portfolio-performance data, and the like and analyzes the gathereddata to make various determinations. One such determination is thecurrent rate of return being achieved on portfolio assets. Otherdeterminations may be measures of portfolio risk, tradeoffs betweenalternative assets, measures of how well strategic goals are being met,and estimates of future performance.

Currently there are many software programs that serve as analyticaltools for the portfolio manager (as well as for the asset manager, fundadministrator, and the like). There are various services, pertinenttimely publications, and other resources for assisting in the foregoinganalyses. Examples include attribution analysis, risk analysis,post-implementation analysis, asset-allocation models, portfoliooptimizers, and the like. These systems and services provide informationregarding how past performance was achieved, or they provide context inwhich to make decisions about new investment activity. What mostdistinguishes these systems from most embodiments of the invention to bedescribed below is that their primary result is the evaluation of past,current, or future performance of specific assets individually. So theyaffect or evaluate what is referred to as strategic decisions: whichspecific assets to buy and which assets to sell. Risk analyzers,portfolio strategizers, and some other existing software systems canprovide simulations of likely portfolio performance, using forecastingmethods that evaluate estimated future outcomes. Other analytic systemsrecast theoretical portfolios in accordance with historical marketinformation and rules for asset selection to evaluate the effectivenessof hypothetical or alternative portfolio-construction strategies (i.e.,methods for selecting assets).

Although such tools are helpful in devising portfolio strategy in arational way, it has been recognized that the strategy thus set and thetrading operations used in response do not alone determine theportfolio's performance. Extensive research and experimentation haveshown that it is also affected by aspects of day-to-day portfoliomanagement that are particularly vulnerable to irrational andnon-optimizing decision-making.

Such decision-making tends to occur, for example, in connection withmoving assets into and out of the portfolio to accommodate investors'contributions and withdrawals. For instance, the objective of sellingoff 5% of a portfolio can be accomplished through any number of tactics.The portfolio manager can sell 5% of each asset position or sell all ormost of a few assets that add up to 5% of the portfolio. The manager canalso choose to reach this objective by selling assets owned the longestor assets whose market value is above or below their purchase price.

The choice of tactics to use in buying assets is similarly broad. Amanager who is trying to increase the portfolio's exposure to aneconomic sector (e.g., biomedical, banking, technology, etc.), forexample, can spread out the purchase across all assets comprising thesector, concentrate the purchase in a few sector assets, or meet thegoal by buying only one asset in the sector. Selection of the approachmay depend on many factors, not all of which are consistent withtraditional economic theory.

The understanding of tactical biases that tend to drive such selectionsis an extension of work done over the past thirty years in the academicarea known as behavioral economics or behavioral finance. Behavioraleconomics is the intersection of psychology and economics. Emotions,ignorance, and faulty cognition work to make human decision making fallshort of the purely rational model. This research indicates that highlytrained, professional investment managers fall prey to these samebiases. Biases routinely drive tactical portfolio-investment decisions,and their impact is, in all likelihood, non-optimizing, largelyunintended, and certainly unexamined or unmeasured. Yet these unexaminedbiases regularly affect the performance of tens of thousands ofprofessionally managed investment portfolios.

Among the types of bias that have been observed is the propensity tofavor selling appreciated assets rather than those that havedepreciated. This preference for selling winners over losers does not,in any systematic way, have much to do with which assets may performbetter in the future. Empirical research instead suggests that this typeof behavior, referred to in economic literature as the “dispositioneffect,” is non-optimizing and may be motivated by the desire toexperience the emotional pleasure of locking in a win while avoiding theunpleasant realization of a loss. A second example of investor biasinvolves the predisposition to buy assets that were previously owned andsold at a gain. Such a bias, referred to as contra-positive investing,is based entirely on subjective factors and reflects no objectivecriteria regarding the familiar asset's expected future performancerelative to less-familiar ones. In both cases the biased decision isneither reflective of a portfolio strategy nor germane to realizingeffective trading.

Although such biases certainly affect the quantities that existingsoftware systems measure, those quantities tend not to be well suited tohelping portfolio managers focus on them or on others of the portfolioperformance's more-tactical aspects.

SUMMARY OF THE INVENTION

We have developed a system that generates performance measuresparticularly sensitive to tactical biases and other aspects ofmanagement tactics.

Our system operates by starting with market data and data that describethe portfolio of interest's holdings at various intervals (typicallydaily) during an observation period. From these inputs it computes thereturns, or other performance measures, of hypothetical portfolios whoseholdings are drawn from the set of assets held by the actual portfolioat some time during the observation period.

This approach differs from other methods of portfolio analysis thatevaluate how results might have varied had different assets been boughtor sold. Although it is conceivable for a performance measure producedby our system to reflect a hypothetical portfolio whose strategy differsmarkedly from the actual portfolio's, those hypothetical portfolios'derivation from the actual portfolio's asset set tends instead to makethe resultant adjusted portfolios largely share that strategy, so thedifferences between the hypothetical portfolios' performances and thatof the actual portfolio tend to be particularly sensitive to tacticalfactors and less sensitive to strategic factors (and, for that matter,to trading effects).

In some embodiments, the system arrives at those hypothetical portfoliosby modulating—e.g., by adding actions to, deleting actions from,increasing and/or decreasing the sizes of, and/or shifting the timingof—the many daily actions (such as buys and sells) that occurred in theactual portfolio.

In any event, one way of using the resultant returns is to generate anindication of a bias's impact. For example, the positions in whichactions of the type that that exhibit (or may exhibit) a bias areclassified in accordance with the attribute that is the candidate biassource; if a bias for or against winners is to be detected, for example,that attribute may be unrealized margin, and positions' segregationbetween winner and loser classes is determined in accordance withwhether their unrealized margins are, say, positive or negative. Thenthe hypothetical portfolio is constructed by so modulating the actualportfolio's actions that the actions of interest in the hypotheticalportfolio occur less disproportionately in those classes than they do inthe actual portfolio; typically, the modulation is performed in such amanner that the relevant actions' occurrences in each class are inproportion to the portfolio positions' memberships in them.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a block diagram of one type of computer system on which thepresent invention's teachings can be implemented.

FIG. 2 is a flow chart of a routine used by the illustrated embodimentto compute a bias measure.

FIG. 3 is a flow chart of a routine for presenting an output indicativeof a bias's impact.

FIGS. 4A and 4B together constitute a flow chart setting forth one ofthe operations for computing alternative portfolio returns.

FIG. 5 depicts one of the illustrated embodiment's output displays thatresults from bias assessment.

FIG. 6 depicts another of the illustrated embodiment's output displaysthat results from bias assessment.

FIG. 7 is one of the illustrated embodiment's output displays thatresults from an assessment of timing advantage.

DETAILED DESCRIPTION OF AN ILLUSTRATIVE EMBODIMENT

Although the present invention can be practiced in virtually any type ofcomputer system, and although data employed by the system will probablybe supplied in many cases from separate, often remote, server systems,it can in principle be implemented in a simple personal computer such asthe one that FIG. 1 depicts. That computer 10 includes among its usualcomponents a microprocessor 12, a RAM 14, and persistent storage, suchas a local disk drive 16, interconnected by a bus 18. For the sake ofsimplicity we will assume that the disk drive not only contains programcode that configures the computer system to perform the analysesdescribed below but also stores investment data that describe one ormore portfolios and include relevant market information. Such programcode may have been loaded onto the disk from a removable storage deviceor through a communications interface 20 from some remote source. Thedata may have been obtained similarly.

Possibly in response to commands entered from, for example, a keyboard22, the system would perform a selected analysis, and it would producean output indicative of the result, possibly sending it out through thecommunications interface 20 or presenting it in human-readable form on adisplay 24 or other output device.

The software configures the computer system as an analysis tool thatanalyzes investment tactics by operations that in many cases includeidentifying and offsetting portfolio actions such as the sales andpurchases of assets. The purposes for such operations may includedetecting tactical biases, assessing their statistical significance,measuring their impacts, and determining how effectively proposedtactical modifications would be in reducing those impacts or otherwiseimproving tactical performance. Such an analysis can be applied toportfolios of any asset type, examples of which include stocks, bonds,options, asset-backed securities, real property, commodities, mutualfinds, Exchange Traded Funds (ETFs), cash or cash-equivalents, andindividual portfolios in a group of portfolios and/or other assets. Itcan be used to evaluate any form of economic interest in financialassets, such as direct ownership, indirect ownership, loans, mortgages,options, short-selling, short-covering, etc. And it is applicable to anyportfolio structure, such as a mutual fund, a hedge fund, a privateportfolio, a partnership, a bank, an insurance company, a pension find,etc.

Among the modes in which the system to be described below can be used toevaluate investment tactics are: a) direct analysis, where tactics aremodified and an output is generated from the resulting adjustedportfolio's performance, b) comparative analysis, where that output insome way compares the adjusted portfolio's performance with the subjectportfolio's, and c) bias analysis, where portfolio assets arepartitioned into two or more groups in accordance with one or moreinvestment attributes, the adjusted portfolio is obtained by modulatingactions on assets in accordance with the groups to which the assetsbelong, and the output is based on the performance of the adjustedportfolio, either directly or in comparison with the subject portfolio.

One of the purposes for which the illustrated embodiment can be used isbias analysis; it can be used to identify and measure the impact of amultitude of biases. Such biases can be based on any attribute orcharacteristic associated with investment decisions. Examples of biasesinclude: a) the gain effect, which is the tendency to sell assets with again or profit disproportionately; b) the age effect, which is thetendency to sell an asset in accordance with its tenure; c) thestop-loss effect, which is selling off assets that drop in valueprecipitously; d) the experience effect, which is the tendency forpurchases and/or sales of a specific asset to be influenced byexperiences with previous holdings in the same asset, and e) theattention effect, which is the tendency to buy and/or sell assets duringperiods of unusually high relative trading volume. (A security's volumeon a given day is the total value of that security that changed hands onthat day in the whole market for that security, whereas the relativetrading volume is the ratio of that value to the average of volume overtime.)

FIG. 2 depicts the routine that the illustrated embodiment uses toproduce a bias measure. As other embodiments will, the illustratedembodiment requires detailed historical portfolio and market data uponwhich to base the analysis, and FIG. 2's block 26 represents obtainingthose data. The portfolio data would either describe the assets bought,sold, and held over some historical period or contain information fromwhich such quantities can be inferred. The length of the historicalperiod will depend on the situation, but, to obtain results that arestatistically meaningful, a duration of, say, at least six months ispreferable.

The system will typically although not necessarily receiveportfolio-specific data separately from data for the market in general.The signals representing the portfolio data can be sent from a keyboardbut will more often come from a communications interface or removablestorage medium. The data will represent the actual portfolio's positionsin respective assets for each of a sequence of market days or otherrecord times. (The data will usually describe a real-world portfolio,but we will refer to portfolios represented by the input data as“actual” portfolios even if no real-world portfolio corresponds tothem.) Often, those data will represent that information compactly. Forexample, the data may in some cases be presented predominantly in termsof trades, so that the absence of an explicit entry for a given stock ona given day represents the fact that the portfolio contains the samenumber of shares, on a split-adjusted basis, as it did after the traderepresented by the last explicit entry for that stock.

Although these data's precise type will depend on the application, letus assume for the sake of concreteness that the portfolio informationavailable to the illustrated embodiment includes, for each asset, foreach reporting observation during the analysis timeframe: a uniqueidentifier (e.g., its CUSIP number); the amount held (which may be givenas numbers of shares, as a dollar or other-currency value, as aweight—i.e., in terms of the percentage of the total portfolio valuethat the asset represents—or in some other convenient measure); adetailed list of all portfolio transactions; net purchases or sales; andnet portfolio income earned other than from asset performance (e.g.,interest on cash, fees from lending assets, cost of margins or shorting,etc.). The data may also specify or enable one to infer net cash flowsin or out for the portfolio for each observation time, although mostanalyses that the illustrated embodiment performs do not require suchinformation.

In principle any observation frequency can be used, such as quarterly,monthly, weekly, daily, hourly, etc., or any combination thereof. Inpractice, daily data will likely be the most typical. For the sake ofconvenience we will therefore refer to successive observations of theportfolio positions' values on a sequence of “days,” but day should beunderstood as a proxy for whatever observation period is convenient.Similarly, we will use the term stock for the sake of convenience torefer to any investment asset, not just those of the equity variety. Wewill also use the term position for the sake of convenience to refer tothe total amount of any investment asset held in a portfolio. Forexample, if stock A has a market value or price of $15.00 and 100 sharesare held in the portfolio, the dollar position of stock A is equal to$1,500 (15.00×100=1,500). The same position expressed in weight termswould be the result of the dollar position divided by the portfolio'stotal dollar value.

The illustrated embodiment next uses the historical portfolio data todetermine the daily weights of each position, as FIG. 2's block 26indicates. Some embodiments may utilize weights provided as part of theportfolio data, and other embodiments may determine them from otherportfolio data. The illustrated embodiment both accepts weights as aninput and determines them from portfolio data stated in other terms. Wewill describe the determination of position weights as part of thisillustration.

The daily weight w(s, i) of the position in an asset s on day i isdetermined as the ratio of the beginning-of-day (“bod”) value for theposition divided by the total portfolio's bod value. So the positionweight for any day i indicates the relative size of the asset within theportfolio prior to any actions (buys, sells), as well as any corporateevents, of the asset on that day. For example, if asset z had a bodposition value of $300 and the total portfolio's bod value for the sameday was $10,000, then the position's weight for that day was 0.03 (300/10,000). Although the preferred embodiment uses weights to analyzeactions that affect positions, other embodiments may implement certainaspects of the current invention by using non-weight representations forpositions and actions.

The quantity defined above as an asset s's weight w(s, i) on day i isthe quantity most intuitively appealing for that purpose; the sum of theweights of all the portfolio's assets is then 1.0 (with shorts andmargin assigned negative weights, as discussed below). Some embodimentsmay include cash as an asset, while other embodiments may exclude cash,focusing exclusively on non-cash assets. But it is possible to use someother quantity that indicates how much of the portfolio the position instock s represents on day i; some embodiments may represent weights asspreads relative to a benchmark, for example, in which case negativeweights would be assigned to any underweighted position, and the sum ofthe weights would be zero. So, if the S&P 500 Index contained 1.5% ofasset X and the subject portfolio contained 1.2% of asset X, therelative weight of the asset would be (0.003) or a negative 0.3%.

To explain how the illustrated embodiment uses these data to assessbias, we will use the variables defined below to refer to the portfoliodata, market data, and quantities that we can derive from them. First,we will adopt the following index variables and definitions:

-   -   s will represent “stocks,” as broadly defined above; the        analysis below can be performed for any collection of assets or        securities, including standard and arbitrary representations of        assets such as sectors, industries, individual asset portfolios        in a set of portfolios and/or other assets, or for any factor        relating to stock returns, including style.    -   S will represent all stocks comprising the relevant universe        throughout the appropriate historical timeframe. Examples of        asset universes include: all bonds issued in North America, all        stocks traded on the Tokyo Exchange, all commodities traded on        the Chicago Mercantile Exchange, or all stocks comprising the        Standard & Poor's 500 Index (S&P 500). The members of any        universe may change over time, with the likelihood of change        increasing as the timeframe lengthens. A universe of particular        interest here is the set of all assets held at one time or        another by the portfolio of interest during some observation        interval.    -   S(i) represents all stocks in the relevant universe on day i,        commonly those stocks comprising the subject portfolio or its        comparative benchmark. So S(i) is a subset of S    -   i, j, k, and l all represent days.    -   p represents a period, i.e., a number of days.

The illustrated embodiment supplements the portfolio data with marketdata. Examples of general market data, which would typically be obtainedseparately from the portfolio data, include the following:

-   -   a. price(s, i) will represent the price of stock s at the close        of day i.    -   b. incomeAmount(s, i) will represent ordinary dividend of        security s with ex-day of i. (If entitlement to the dividend is        determined, as is typical, by ownership of the security at the        end of a given day, the following day is that security's        ex-day).    -   c. eventAmount(s, i) will represent special dividend or proceeds        from spin-off on ex-day i.    -   d. splitFactor(s, i) will represent the split factor on ex-day        i.    -   e. volume(s, i) will represent the volume of market transactions        on day i.

Those skilled in the art will be familiar with the use of these marketdata in determining the return of a stock for any day i and over anyperiod of time (i, j). Examples of other fundamental data that someembodiments may use instead or in addition are earnings per share, bookvalue per share, short-term forecasted earnings-growth rate, long-termforecasted earning-growth rate, etc. Among the analyses that theillustrated embodiment can perform is, as was mentioned above, biasdetection. The biases that the illustrated embodiment will be used todetect are biases in taking what we call investment “actions,” and FIG.2's block 28 represents identifying actions by computing “action values”from the portfolio and market data. Before we discuss how it computesthose values, though, we will consider what kinds of actions are ofinterest.

A given position's action value will be taken to be some measure of thedegree to which a position differs from what it would have been in theabsence of the portfolio manager's actions. Among systems that usevarious aspects of this application's teachings, there will likely besome variation in just what types of differences are considered for thispurpose. Some such systems, for instance, may consider differences in aposition's dollar value or differences in numbers of shares. Therefore,such systems will consider an action of some magnitude to have occurredin any position to which the manager has added additional shares.

But the types of differences that are considered by the embodiment onwhich we will principally concentrate are not dollar or sharedifferences; they are weight differences, and most of the quantities tobe dealt with below are based on weight. It will therefore be helpful tomention a couple of the implications of operating in the weight domain.One is that a position is expressed in the weight domain as the ratio ofits dollar (or other-currency) value to that of the portfolio as awhole. So, whereas the total of all of a portfolio's positions in dollaror share terms may change from day to day, the total in weight termsalways equals one.

The other implication is that weight-based action values differ fromdollar- or share-based action values not only in the terms used toexpress them but also in whether an action is considered to occur atall. For example, consider a day on which a portfolio manager receives alarge infusion of funds and promptly so invests them that the dollarvalues of all his positions end up being double what they would havebeen if no such transactions had occurred. A dollar-valued actionmeasure will reflect significant actions in all those positions. But theweights are all what they would have been in the absence of thosetransactions. As we explain below, we call such weight values thepositions' “no-action weights,” and we treat a position's action valuein the weight domain as the difference between its no-action and actualweights. So a weight-based action assessment in this scenario would findno action in any position even though a dollar-based assessment wouldfind actions in all of them.

We now turn in detail to the various quantities that the illustratedembodiment uses in making such assessments. To compute action values asdepicted in FIG. 2's block 28, the illustrated embodiment relies uponthe following definitions:

-   -   r(s, i) is the return of security s on day i. Asset returns are        derived from the market information described above. There may        be some variation in what various embodiments consider to be        included in return; those that the illustrated embodiment        includes will be identified in due course.    -   r(s, i, j) is the return of the security s from the end of day        i−1to the end of day j:

${r\left( {s,i,j} \right)} = {{\prod\limits_{{k = i},j}^{\;}\;\left( {1 + {r\left( {s,k} \right)}} \right)} - 1}$

-   -   c(s, i) The contribution c(s, i) on day i of a position in asset        s is the measure of the proportion of the portfolio's return for        that day that the asset-s position contributed. That        contribution is the product of asset s's position weight and its        return:        c(s,i)=w(s,i)·r(s,i)    -   So contribution is positive when a gain occurs and negative when        a loss occurs.    -   pr(i) is portfolio return or the return for all assets s held        for each day i:

${{pr}(i)} = {\sum\limits_{s \in {S{(i)}}}^{\;}\;{c\left( {s,i} \right)}}$

-   -   pr(i, j) is portfolio return from the end of day i−1to the end        of day j:

${{pr}\left( {i,j} \right)} = {{\prod\limits_{{k = i},j}^{\;}\;\left( {1 + {{pr}(k)}} \right)} - 1}$Action Identification

In the illustrated embodiment an action is something that tends to bethe result of decisions made by portfolio managers; as was mentionedabove and will be explained in more detail presently, an action is adifference between the actual weight and the weight that would result ifthe portfolio were affected only by returns. Although not all actionsunder this definition map one-to-one to dollar-value purchases andsales, we will characterize all such actions on long positions as eitherbuys or sells, whereas actions on short positions will all becharacterized as shorts or covers. For convenience we will focus thisillustration on “buying” and “selling” long positions.

To discern an action based on weights we begin by calculating an asset'sno-action weight. In the illustrated embodiment, the no-action weightnaw(s, i) for an asset s on day i is based on that asset's return r(s,i−1) from the prior day relative to the entire portfolio's returnpr(i−1) on the prior day:

${{naw}\left( {s,i} \right)} = {\frac{1 + {r\left( {s,{i - 1}} \right)}}{1 + {{pr}\left( {s,{i - 1}} \right)}}{w\left( {s,{i - 1}} \right)}}$

The no-action weight is the weight that an asset that would be expectedto have on day i if no transactions occurred on the prior day, and aposition is considered to have been subject to an action on the previousday only if its weight differs from its no-action weight. The no-actionweight moves proportionally to the return of the stock (s) relative tothe return of the portfolio (S). For example, if assets x and z were theonly assets in a portfolio, they are equally weighted on day i−1, andtheir respective returns for day i 1 were 2% and 0%, then the no-actionweight naw(x, i) of asset x on day i equals(1.02×0.50)/(1.02×0.50+1.00×0.50)=50.5%, while asset y's is (1.00×0.50)(1.02×0.50+1.00×0.50)=49.5%.

Having determined both the daily weight and the no-action weight foreach asset s's position, the system performs the operation that block 30represents: it computes the weight action(s, i) of an action in asset son day i as the weight of the asset on day i+1 minus the no-actionweight on day i+1:

$\begin{matrix}{{{action}\left( {s,i} \right)} = {\Delta\;{w\left( {s,i} \right)}}} \\{= {{w\left( {s,{i + 1}} \right)} - {{naw}\left( {s,{i + 1}} \right)}}}\end{matrix}$

A typical analysis (bias or otherwise) may concern itself with onlycertain components of the actions, i.e., only with action componentsthat meet certain action criteria. For example, a given analysis mayconcern itself only with actions that decrease a position's weight(e.g., a sell for a long position). A further criterion may be that theaction occur in a given type of position. For instance, most analyseswill restrict their attention to actions that occur in assets other thancash. The block-30 operation therefore includes restricting the actionsto those that are relevant to the particular analysis currently beingperformed. This often involves classifying the actions in accordancewith one or more of those criteria. To this end the illustratedembodiment divides actions into components—e.g., into sells and buys ina long-position-only portfolio—to which it assigns corresponding values.

To understand how the illustrated embodiment arrives at these values,consider sells. In a long-position-only portfolio, an action is a sellonly if the action's value (i.e., the amount by which the next day'sweight is algebraically greater than the no-action weight) is negative:

${{sell}\left( {s,i} \right)} = \left\{ \begin{matrix}{{- \Delta}\;{w\left( {s,i} \right)}} & {{{if}\mspace{14mu}\Delta\;{w\left( {s,i} \right)}} < 0} \\0 & {otherwise}\end{matrix} \right.$

A buy action is defined similarly:

${{buy}\left( {s,i} \right)} = \left\{ \begin{matrix}{\Delta\;{w\left( {s,i} \right)}} & {{{if}\mspace{14mu}\Delta\;{w\left( {s,i} \right)}} > 0} \\0 & {otherwise}\end{matrix} \right.$

Sometimes it is useful to understand the level of selling or buying thatis occurring within the portfolio as a whole. The illustrated embodimentuses the above-defined sell value to compute the sum sales(i) of everysell(s, i) of an asset s in the portfolio on day i:

${{sales}(i)} = {\sum\limits_{s \in {S{(i)}}}^{\;}\;{{sell}\left( {s,i} \right)}}$

From that value the turnover of portfolio from day i to the end of day jcan be determined by summing sells over the relevant time period:

${{turns}\left( {i,j} \right)} = {\sum\limits_{k = i}^{\; j}\;{{sales}(k)}}$

The turns value can be annualized by using the following expression:

${{{turnover}\left( {i,j} \right)} = {\frac{\sum\limits_{k = i}^{\; j}\;{{sales}(k)}}{j - i + 1} \cdot 252}},$where 252 is the average number of trading days in a year.Determining Asset Attributes

Certain analyses, such as measuring the bias of selling winnersdisproportionately to losers, require selecting or sorting assets byvarious financial attributes or characteristics. In many analyses theattribute will be a quantity introduced in the following discussion, orit may be a combination of them or some different quantity.

Selling an asset can involve selling all or a portion of a position. Theproportion of asset s's position sold on day i is referred to asfraction realized fr(s, i), which is calculated as the ratio of thatasset's sell on day i divided by its no-action weight on the next day:

${{fr}\left( {s,i} \right)} = \frac{{sell}\left( {s,i} \right)}{{naw}\left( {i + 1} \right)}$

When 100% of an asset is sold on day i, its no-action weight on day i+1will be the same as its previous day's sell value, and the resultingfraction realized will equal 1.0. When less than all of the assetposition is sold on day i, its no-action weight on day i+1 will begreater than the value of sell(s, i), and the resulting fractionrealized will be between zero and 1.0.

Buying an asset can involve adding a new position or adding to anexisting position. The proportion fp(s, i) of the position in stock sbought on day i is calculated as the ratio of that asset's buy on day ito its weight on the next day:

${{fp}\left( {s,i} \right)} = \frac{{buy}\left( {s,i} \right)}{w\left( {i + 1} \right)}$

Since the position weight is calculated as of the beginning of each day,any buy on day i that initially establishes the portfolio's position ina given asset will have the same value as that position's weight on dayi+1, and the resultant fraction purchased will equal 1.0. Forincremental purchases the amount purchased on day i will be less thanthe position's weight on day i+1, with the resulting fraction purchasedbeing between zero and 1.0.

In some embodiments it will be useful to consider the portion of aposition that was not sold. This fraction unrealized fu(s, i) iscalculated as 1 minus the fraction realized:fu(s,i)=1−fr(s,i)

In determining certain bias impacts it is useful to measure portfoliopositions' profitabilities (gains/losses). One common measure ofprofitability is unrealized margin. Unrealized margin is the ratio ofunrealized profits to cost basis. To help define margin consistentlywith the illustrated embodiment's weight scheme, we define a series ofintermediate quantities.

Whereas many of the quantities defined so far have been stated in weightterms, i.e., in how many dollars of that quantity there are on a givenday for every dollar of total portfolio value on that day, it will beconvenient instead to define certain other quantities in what we call“unit-dollar” terms, which is the number of dollars of some quantity ona given day for each dollar of the total portfolio value on day 0 ratherthan for each dollar of the total portfolio value on the given day.

One example of such a quantity is unrealized contribution. For a givenasset s, unrealized contribution uc(s, i, j) tells how many dollars ofthe day-j portfolio value were contributed by the return on that assetsince some day i for each dollar of the portfolio's value on day 0. Ourdiscussion of that value will use two interim terms: scaling factor andgain.

The scaling factor sf(i) indicates how many dollars of portfolio valueon day i there are for every dollar of portfolio value on day 0:sf(i)=1+pr(0,i−1)

It is used to compute the dollar unit value unitValue(s, i) of asset son day i:unitValue(s,i)=w(s,i)·sf(i),which indicates how many dollars of asset s there are on day i for everydollar of day-0 portfolio value.

A stock s's day-i gain g(s, i) is the product of its contribution c(s,i) and the scaling factor sf(i):g(s,i)=sf(i)·c(s,i).

Having defined these quantities, we can define an asset s's day-junrealized contribution uc(s, i, j) since day i. What we want is adollar-equivalent quantity that represents only that portion of aposition's contribution that has not come out of the position by sales.To that end, we assume that each partial sale's proceeds come from allof that position's components proportionately; a sale on day j ofone-third of a position—i.e., a sale that leaves a fraction unrealizedof two-thirds—leaves behind two-thirds of day j−1's unrealizedcontribution, two thirds of day j's additional gain, and two-thirds ofevery other component of the position, such as its cost. So we cancompute uc(s, i, j) inductively as the product of day j's fractionunrealized fu(s, j) and the sum of the previous day's unrealizedcontribution uc(s, i, j−1) and day j's (dollar-equivalent) gain g(s, j):uc(s,i,j)=fu(s,j)·[uc(s,i,j−1)+g(s,j)]

For example, suppose that day j is the day on which the portfolio'sposition in an asset s is first established and that no sales occur onday j. Since no sales occurred on day j, the fraction unrealized isunity. Therefore, that position's day-j unrealized contribution uc(s, i,j) since any previous day i will equal the sum of (a) previous day'sunrealized contribution, which is zero because the asset position didnot exist then, and (b) the gain that occurred on day j. That is, theunrealized contribution is simply the gain from day j. Since gain is adollar equivalent, so is unrealized contribution.

An asset s's unrealized margin um(s, i, j) is a measure of the degree towhich that asset has been a winner or loser in the interval from day ito day j. It can be computed from unrealized contribution as the ratioof that quantity to the amount by which asset s's day-j unit valueexceeds that quantity:

${{um}\left( {s,i,j} \right)} = \frac{{uc}\left( {s,i,j} \right)}{{{unitValue}\left( {s,j} \right)} - {{uc}\left( {s,i,j} \right)}}$

Related values for realized margin and margin as a whole can be definedwith the aid of a few additional definitions

The unit-dollar quantity that represents the contributions since day ithat are realized from a sale of stock s on day j is:ra(s,i,j)=fr(s,j)·[uc(s,i,j−1)+c(s,j)·(1+pr(0,j−1))]

Of the contributions from asset s since day i, the unit-dollar quantityfor the total realized during the period from j to k is given by:

${{rc}\left( {s,i,j,k} \right)} = {\sum\limits_{l = j}^{k}\;{{ra}\left( {s,i,l} \right)}}$

The unit-dollar quantity for total sales from day j through day k is:

${{proceeds}\left( {s,j,k} \right)} = {\sum\limits_{l = j}^{k}\;{{{sell}\left( {s,l} \right)} \cdot \left( {1 + {{pr}\left( {0,{l - 1}} \right)}} \right)}}$

These quantities enable us to define a quantity realizedMargin(s, i, j,k) that represents what the profit margin in the asset-s position isover the period from day j to day k if the basis is taken as thatposition's value on day i:

${{realizedMargin}\left( {s,i,j,k} \right)} = \frac{{rc}\left( {s,i,j,k} \right)}{{{proceeds}\left( {s,j,k} \right)} - {{rc}\left( {s,i,j,k} \right)}}$

The total margin, which is a combination of the realized and unrealizedmargins, is given by:

$\quad{{{margin}\left( {s,i,j,k} \right)} = \frac{{{rc}\left( {s,i,j,k} \right)} + {{uc}\left( {s,j,k} \right)}}{{{unitValue}\left( {s,k} \right)} + {{proceeds}\left( {s,j,k} \right)} - {{rc}\left( {s,i,j,k} \right)} + {{uc}\left( {s,j,k} \right)}}}$

Among other values used in partitioning assets is age. An asset s's ageage(s, i) on day i is the average time it has been held in the portfolioup to day i, weighted by interim purchases:

${{age}\left( {s,i} \right)} = \left\{ \begin{matrix}{1 + {\left( {1 - {{fp}\left( {s,i} \right)}} \right) \cdot {{age}\left( {s,{i - 1}} \right)}}} & {{{if}\mspace{14mu}{w\left( {s,i} \right)}} \neq 0} \\0 & {{{if}\mspace{14mu}{w\left( {s,i} \right)}} = 0}\end{matrix} \right.$Thus, for two assets y and z both held in the portfolio for the samenumber of days, asset y will have a lower age if it experienced interimbuys and asset z did not.

A complementary quantity life(s, i) is the average time that an asset swill remain in the portfolio from day i until the point at which it isfully liquidated—i.e., until w(s, i)=0. The measure used by theillustrated embodiment takes interim partial sales into account:

${{life}\left( {s,i} \right)} = \left\{ \begin{matrix}{1 + {\left( {1 - {{fr}\left( {s,i} \right)}} \right) \cdot {{life}\left( {s,{i + 1}} \right)}}} & {{{if}\mspace{14mu}{w\left( {s,i} \right)}} \neq 0} \\0 & {{{if}\mspace{14mu}{w\left( {s,i} \right)}} = 0}\end{matrix} \right.$Thus, for two assets y and z, both of which will be held in theportfolio for the same number of days, asset y's life will be less if itwill experience interim partial sales and asset z will not.

If no sells occur in the asset-s position until after day j, the costvalue cost(s, i, j) is its weight on day i plus the sum of all buyactions that occur from that day through day j−1. If sells occur duringthat interval, they reduce the cost proportionally:

${{cost}\left( {s,i,j} \right)} = \frac{{{w\left( {s,j} \right)} \cdot \left\lbrack {1 + {{pr}\left( {0,{j - 1}} \right)}} \right\rbrack} - {{uc}\left( {s,i,j} \right)}}{1 + {{pr}\left( {0,{j - 1}} \right)}}$We will also use the notation cost(s, j)≡cost(s, 0, j).Bias Assessment

With these quantities defined, we are ready to describe how theillustrated embodiment implements bias assessment. As blocks 32, 34, and36 indicate, the general approach is to compute values of the attribute(such as unrealized margin) on which the bias to be measured is based,divide the positions into classes (such as winners and losers) inaccordance with those attribute values, and compute the bias measurefrom those groups. The particular attribute on which the classificationis based depends on the particular bias of interest.

Sometimes the partitioning is based upon a sort value. For example,unrealized margin can be used as the sort value to define winners as theassets whose unrealized margins are positive and losers are defined asthose whose unrealized margins are negative. In that example, there is afixed threshold, zero, that divides the groups: all positions whoseunrealized margins exceed zero fall into the winners group, while thosewhose unrealized margins are less than zero belong to the losers group.

When the threshold is fixed, the resultant groups' sizes are not alwaysthe same. But the system may provide the additional capability ofpartitioning each group in such a manner that each group has the sameweight or number of items. We refer to such groups as “tiles.” If thisapproach is used with unrealized margin as the sort value, partitioninginto two tiles would result in one tile containing the items with thelower unrealized margins and a second tile containing the items with thehigher unrealized margins.

For some purposes, there may be more than two groups into which theitems are to be partitioned. If the groups are not to be tiles in suchcases, then more than one threshold is needed.

With the groups having thus been identified, the bias can be calculated.Each bias will be calculated over an analysis period (starting on day jand up to and including day k). The bias will be calculated as thedifference between two weighted averages. The items comprising theaverage are the cross product of day and security. In other words, foreach day in the analysis period and for each relevant security therewill be one item. (In most analyses, cash will not be considered arelevant security for this purpose, and some analyses may exclude othertypes of assets as well.) Notionally, the item will be represented as(s, l), where s is the stock and l is the day.

For buys, the illustrated embodiment computes bias as the differencebetween the groups' weighted averages of the fraction purchased:

${{bias}_{buys} = {\frac{\sum\limits_{{({s,l})} \in {I\; 1}}{{{fp}\left( {s,l} \right)} \cdot {w\left( {s,l} \right)}}}{\sum\limits_{{({s,l})} \in {I\; 1}}{w\left( {s,l} \right)}} - \frac{\sum\limits_{{({s,l})} \in {I\; 2}}{{{fp}\left( {s,l} \right)} \cdot {w\left( {s,l} \right)}}}{\sum\limits_{{({s,l})} \in {I\; 2}}{w\left( {s,l} \right)}}}},$where I1 and I2 represent the classes into which the items have beenpartitioned. For sells, the bias is the difference between the weightedaverages of the fraction realized:

${bias}_{sells} = {\frac{\sum\limits_{{({s,l})} \in {I\; 1}}{{{fp}\left( {s,l} \right)} \cdot {{naw}\left( {s,l} \right)}}}{\sum\limits_{{({s,l})} \in {I\; 1}}{{naw}\left( {s,l} \right)}} - {\frac{\sum\limits_{{({s,l})} \in {I\; 2}}{{{fr}\left( {s,l} \right)} \cdot {{naw}\left( {s,l} \right)}}}{\sum\limits_{{({s,l})} \in {I\; 2}}{{naw}\left( {s,l} \right)}}.}}$

Although the illustrated embodiment uses weight and no-action weight asthe values by which to weight the items, some embodiments may use otherweighting schemes; some may be based on cost, for example. Indeed, somemay use dollar amounts rather than ratios.

Different embodiments will likely give the user different choices amongattributes on which to base the classification. If the purpose is toassess the “gain effect,” the attribute of interest is unrealizedmargin. Some may distinguish among long- and short-term gain effects.For long-term gain effect, the specific unrealized-margin quantity maybe the unrealized margin unrealizedMargin(s, 0, l) since portfolioinception, while short-term gain may be some value such as theunrealized margin unrealized margin(s, l−-p, l) in the last p days,where typical values of p are 21, 42, and 63.

Examples of attributes that can be used for classification when the biasto be assessed is the “experience effect” are the above-defined realizedmargin and total margin. This effect, too, can be assessed on bothshort- and long-term bases.

If the bias to be detected is the age effect, the attribute on which theitems are sorted is the asset's age age(s, l).

Another effect for which the system may test is the “momentum effect.”In this case, the attribute is “momentum.” An asset s's p-day momentummomentum(s, i, p) on day i is its cumulative return between andincluding days i and i−p+1:momentum(s,i,p)=r(s,i−p+1,i)

The parameter p determines whether the effect detected is the short- orlong-term momentum effect. Typical values for long- and short-termmomentum effects are p=252 and p=21, respectively.

A related effect is the “aged-momentum” effect, for which sorting isbased on the momentum at the time of purchase. Now, a position may beestablished over multiple days, so the value for that quantity should bea weighted average of the various purchases' contributions. The fractionpurchased fp(s, i) is a convenient value to use for this purpose; on theday after the first purchase, the fraction-purchased value is unity, itis zero on days that follow no purchases, and it is some non-zerofraction on the days that follow subsequent purchases. So asset s'sp-day aged momentum on day i can be calculated as follows:

agedMomentum(s, i, p) = [1 − fp(s, i)] ⋅ agedMomentum(s, i − 1, p) ⋅ +fp(s, i) ⋅ momentum(s, i, p)

Another effect that some embodiments may be arranged to detect is a biasbased on the assets' relative volumes. An asset s's p-day relativevolume relativeVolume(s, i, p) on day i is the ratio its dollar volumeon day i bears to the average daily dollar volume of s between andincluding days i−1 and i−p:

${{relativeVolume}\left( {s,i,p} \right)} = {\frac{{{price}\left( {s,i} \right)} \cdot {{volume}\left( {s,i} \right)}}{\sum\limits_{j = {i - p}}^{i - 1}\;{{{price}\left( {s,j} \right)} \cdot {{volume}\left( {s,j} \right)}}}.}$

The system could similarly base bias detection on an aged version ofthis quantity:

ageRelativeVolume(s, i, p) = [1 − fp(s, i)] ⋅ agedRelativeVolume(s, i − 1, p) ⋅ +fp(s, i) ⋅ momentum(s, i, p)

Two further examples of attributes that could be used for classificationare volatility and aged volatility. The p-day volatility of an asset son day i is the standard deviation of daily returns for stock s betweenand including days i−p+1 and i:

${{{volatility}\left( {s,i,p} \right)} = \sqrt{\frac{1}{p - 1} \cdot {\sum\limits_{j = {i - p - 1}}^{i}\;{\left( {{r\left( {s,j} \right)} - \overset{\_}{r(s)}} \right)2}}}},{{where}\text{:}}$${\overset{\_}{r(s)} = {\frac{1}{p} \cdot {\sum\limits_{j = {i - p + 1}}^{i}\;{r\left( {s,j} \right)}}}},$

The aged version of that quantity, i.e., the asset's volatility when itwas acquired, is given by:

ageVolatility(s, i, p) = [1 − fp(s, i)] ⋅ agedVolatility(s, i, 1, p) ⋅ +fp(s, i) ⋅ volatility(s, i, p)

Of course, sorting can be based on other attributes, too.

Statistical Significance

In addition to calculating the bias measure, some embodiments mayadditionally give the user an indication of how statisticallysignificant the bias measure is. One approach to providing thatindication is to compute the t-Statistic adapted for weighted averages.As defined and published by Goldberg, the t-Statistic for a weightedaverage is calculated as follows:

${tStat} = \frac{\overset{\_}{X} - \overset{\_}{Y}}{\sqrt{\frac{\frac{Sx}{\alpha\; x} + \frac{Sy}{\alpha\; y}}{n + m - 2}}\sqrt{\alpha_{x} + \alpha_{y}}}$where n is the number of buys in I1 and m is the number of buys in I2,and where:

$S_{x} = {\sum\limits_{{({s,l})} \in {I\; 1}}^{\;}\;{{w\left( {s,l} \right)} \cdot \left( {{{fp}\left( {s,l} \right)} - \overset{\_}{X}} \right)^{2}}}$$S_{y} = {\sum\limits_{{({s,l})} \in {I\; 2}}^{\;}\;{{w\left( {s,l} \right)} \cdot \left( {{{fp}\left( {s,l} \right)} - \overset{\_}{Y}} \right)^{2}}}$$\alpha_{x} = \frac{\sum\limits_{{({s,l})} \in {I\; 1}}^{\;}\;{w\left( {s,l} \right)}}{n}$$\alpha_{y} = \frac{\sum\limits_{{({s,l})} \in {I\; 2}}^{\;}\;{w\left( {s,l} \right)}}{n}$

Below are the definitions of terms in the above t-Statistic equationwhen calculating for buys:

$\overset{\_}{X} = \frac{\sum\limits_{{({s,l})} \in {I\; 1}}^{\;}\;{{{fp}\left( {s,l} \right)} \cdot {w\left( {s,l} \right)}}}{\sum\limits_{{({s,l})} \in {I\; 1}}^{\;}\;{w\left( {s,l} \right)}}$${\overset{\_}{Y} = \frac{\sum\limits_{{({s,l})} \in {I\; 2}}^{\;}\;{{{fp}\left( {s,l} \right)} \cdot {w\left( {s,l} \right)}}}{\sum\limits_{{({s,l})} \in {I\; 2}}^{\;}\;{w\left( {s,l} \right)}}},$

The definitions for sells are the same, except fp(s, l) is replaced withfr(s, l), and w(s, l) is replaced with naw(s, l).

Such biases and indications of those biases' statistical significanceare performance measures, but so are other quantities described above,as are further quantities to be described below. Such quantities includethose used as bases for classifying positions to detect bias. So acomputer system that embodies the present invention's teachings may givethe user the option of having it generate outputs that represent one ormore of those quantities, and/or quantities based on them, separatelyfrom their use as possible bases for bias. Also, although theillustrated embodiment bases computation of such quantities on weightsand weight-based action values, other embodiments may compute them inways that are separately sensitive to the positions' currency orother-unit values, such as share values. In some embodiments, forexample, a portfolio in which every position's value doubles in sizedespite the absence of any earnings would be considered to have anaction in every position, although the illustrated embodiment would findno action in any of those positions.

Impact

Separate from the question of whether the level of detected bias issignificant statistically is that of what impact the bias has had on theportfolio. The illustrated embodiment treats certain user inputs asrequesting that it present the user some indication of what that impactis. For example, the user may want to know whether the portfoliomanager's bias has made the portfolio's returns more or less than theywould have been in the absence of that bias. The system may give theresults in terms of differences in simple return per dollar of portfoliovalue, or of something more complicated, such as risk-adjusted return.Or, independently of return, the user may want to know what the effectwould be on portfolio risk or on some other figure of merit commonlyapplied to asset portfolios.

The approach to assessing impact is essentially independent of the biasdetected, so no generality will be lost by our concentrating thediscussion below on a bias toward buying or selling winners or losersdisproportionately. Analysis tools that implement this aspect of theinvention will compute the return (or other figure of merit) of aportfolio that is derived from the actual portfolio but has beenadjusted to exhibit less bias. (Typically, that lower bias will be equalor near to zero). The bias's impact is then determined from a comparisonbetween that adjusted portfolio's result and either the actual portfolioitself or a portfolio that is derived from that portfolio and exhibits asimilar bias.

Various of the present invention's embodiments will adopt differentapproaches to determining such adjusted portfolios and making thecomparisons between biased and unbiased portfolios. One approach, forexample, is to determine the unbiased portfolio by adjusting the sizesof the actual portfolio's relevant actions. If the impact to bedetermined is that of a bias toward selling winners, for example, theadjusted portfolio can be the same as the actual portfolio with theexception that the size of each sale of a winner is reduced, and thesize of each sale of a loser is increased, to an extent that removes thebias. That portfolio's return could then be compared with the return ofthe actual portfolio or with the return of a hypothetical portfoliowhose assets are the same and are similarly biased.

The illustrated embodiment adopts a different approach, though. Ratherthan modulate action sizes, the illustrated embodiment adjusts thepositions' holding periods. If the portfolio exhibits a bias towardselling winners, for example, the illustrated embodiment's approach isto increase the winners' holding periods and reduce the losers'.

FIG. 3 illustrates this approach. That drawing's block 60 representsdetermining the average ages of the (typically, two) classes that resultfrom classifying the portfolio's positions in accordance with unrealizedmargin (or some other attribute of interest). Some embodiments maycompute the average by weighting various contributions in accordancewith position weight. But the quantity that the illustrated embodimentuses in weighting contributions to respective average ages for allwinners and for all losers is cost:

${{age}_{I\; 1} = \frac{\sum\limits_{{({s,l})} \in {I\; 1}}{{{age}\left( {s,l} \right)} \cdot {{cost}\left( {s,l} \right)}}}{\sum\limits_{{({s,l})} \in {I\; 1}}{{cost}\left( {s,l} \right)}}};{{age}_{I\; 2} = \frac{\sum\limits_{{({s,l})} \in {I\; 2}}{{{age}\left( {s,l} \right)} \cdot {{cost}\left( {s,l} \right)}}}{\sum\limits_{{({s,l})} \in {I\; 2}}{{cost}\left( {s,l} \right)}}}$

As block 60 also indicates, the illustrated embodiment additionallycomputes not only the respective groups' average ages but also theaverage age of the portfolio as a whole. The reason for the latteroperation is that, rather than comparing the adjusted portfolio'sresults with those of the actual portfolio, the illustrated embodimentcompares it with a hypothetical portfolio in which the positions are thesame as those of the actual portfolio but in which the holding periods,as will now be explained, have all been so expanded or reduced that theyare approximately the same.

Of course, some embodiments will, as was mentioned above, make thecomparison with the actual portfolio's return rather than that of ahypothetical one. But the illustrated embodiment's purpose in generatingan impact measure is more to indicate what effect the detected biastends generally to have than to indicate what it had in the particulartime period on which the computation was based, and making all of thepositions' holding periods equal tends to eliminate anysituation-specific considerations that may have affected the actualportfolio's results. So the illustrated embodiment computes thisunbiased adjusted portfolio, as block 62 indicates, in a manner thatwill in due course be described in connection with FIG. 4.

Before we consider the FIG. 4 routine, though, we will describe atechnique that it uses in arriving at the hypothetical portfolios'returns. In accordance with that technique, it begins with the actualportfolio's return and adjusts it for the effects of certain additionalactions. In many cases, these additional actions are intended to cancelactions that occurred in the actual portfolio, and the bulk of thediscussion below will be couched in terms of offsets to actions thatoccurred in the actual portfolio. In the case of some “offsets,” though,the purpose may be to cause the alternate portfolio to reflect theresults of an action that did not occur in the actual portfolio ratherthan to offset one that did.

Now, the number of actions to be compensated for at the same time maywell be large. Also, the number of different sets of such compensationsto be considered concurrently may itself be large, and each of thosesets may need to be computed for a large number of days. Moreover, theeffect of offsetting a given action interacts with the effects ofoffsetting all the other actions to be compensated for at the same time,so the complexity of computing such compensation could potentially growquite fast with the number of compensated-for actions. But theillustrated embodiment employs a way of computing such compensation thatscales well; its complexity grows only linearly with the number ofactions to be offset.

This approach to computing the alternate portfolio's return for a givenday involves performing a sequence of operations, each of whichimplements an “offset” o. Initially, we will discuss each offset asbeing associated with a respective action or group of actions that haveoccurred in the actual portfolio but whose effects the hypotheticalportfolio's return should not reflect. The operation that is used toimplement a given offset in computing a given day'shypothetical-portfolio return adds that offset's effects to those ofother offsets in such a way that each operation's complexity does notdepend on the number of actions for which compensation is therebyprovided.

One way of carrying out this approach involves computing for each offseto for each relevant day i a respective portfolio-weight value w(o, i)and a corresponding offsetting return value or(o, i) such that thereturn r(o, i) that the actual portfolio would have had in the absenceof the actions to be offset by operation o equals the sum of thatoffsetting return and the product of that weight and the actual return:r(o,i)=or(o,i)+w(o,i)·pr(i).

(Note that here we are overloading the function name w: w(o, i), inwhich the first argument is of the offset-operation type, returns aportfolio weight, whereas w(s, i), in which the first argument is of theasset type, returns the weight of an asset in a portfolio.)

It will sometimes be convenient to think of the offsetting return valueor(o, i) as the result of complementarily weighting the return d(o, i)of a hypothetical offset portfolio, in which case the expression for theadjusted portfolio's return becomes:r(o,i)=w(o,i)·pr(i)+[1−w(o,i)]·d(o,i),where, if the offset o offsets only a single action, on an asset s, d(o,i) is the return of a portfolio 100% invested in s: d(o, i)=r(s, i). Foran offset of a set of actions, d(o, i) could be derived, in a mannerthat will be explained below, from the returns of the assets on whichthe actions in the set occurred.

The advantage of this incremental-computation approach is, as willshortly be seen, that the complexity of adding an offset for a givenaction to that of other actions is independent of the number of otheractions; without using any quantity specific to any action a₁, a₂, . . ., or a_(N-1), the weight and offset-return-contribution values for anoffset that compensates for actions a₁, a₂, . . . , a_(N) can becomputed simply from the corresponding two values for the offset thatcompensates for actions a₁, a₂, . . . , and a_(N-1). As will also beseen below, the turnover that would be required if the adjustedportfolio were actually to be implemented can be computed in a similarlyincremental and scalable manner.

To explain how we arrive at a weight value w(o, i) that implements thisapproach appropriately, we start by considering how to offset a singlebuy action that has occurred in the actual portfolio. We assume that thebuy action occurs on day i−1, and we will determine the value w(o, i)that the weight assumes on day i. Once that value has been determined,the weight value on each subsequent day j, which will increase ordecrease in accordance with the actual and substitute portfolios'relative performances, can be determined from the previous day's weight:

${w\left( {o,j} \right)} = {{w\left( {o,{j - 1}} \right)} \cdot {\frac{1 + {{pr}\left( {j - 1} \right)}}{1 + {r\left( {o,{j - 1}} \right)}}.}}$

From the second equation above for r(o, i) and the fact its “difference”value d(o, i) in the case of a single buy action on a position s is thatposition's return r(s, i), it is apparent that, to roll back the effectof the buy, the weight factor 1−w(o, i) by which position s's return ismultiplied must be negative: the hypothetical alternate portfolio musthave a short position in that asset. In the case of a buy, therefore,w(o, i) must be some value greater than one.

To determine just what that value should be, it is important first toconsider precisely what the nature is of the “actions” that we definedabove. To this end consider a portfolio whose holdings are limited topositions in assets x, y, and z, and let us suppose that the position inx ends up with a weight of 40% rather than its no-action weight of 20%:it had a 20% action (which we refer to as a “buy”). Now, that 20% ofweight could have all come from the position in y, it could have allcome from the position in z, or it could have come in some proportionfrom both. When we offset that buy, though, we want to arrive at theoffset quantities in a way that is independent of where the funds tomake the buy came from. So, as we will demonstrate presently, we model abuy as a purchase made with funds that come from outside the portfolio,i.e., as an infusion of funds into the portfolio. We similarly model asell as a (complete or partial) liquidation of a position and awithdrawal of the liquidation proceeds from the fund.

This can be comprehended best by venturing briefly into the dollardomain. Consider the portfolio as having a value of $100 at the start ofthe day after the buy. According to our model, the 20% action in x meansthat the x position resulted from adding $20 to that position. And,although the funds for adding to that position may have been generatedby liquidating other positions, we model the buy as those funds' havingcome from outside the portfolio. We do this because the liquidations ofother positions are themselves actions, and we are interested here inoffsetting only the position-x action, not in offsetting the others. So,to offset the x-position action by itself, we take $20 out of the xposition—changing it to $20—without adding that $20 to any of the otherpositions. In the dollar domain, that is, offsetting that buy withoutoffsetting any other actions results in the x position's having a valueof $40−$20=$20 out of a total portfolio value of $100−$20=$80.

We now return to the weight domain and observe that the offset's resultis a position-x weight of ($40−$20)/($100−$20)=25%. More generally, thatis, the day-i weight w′(s, i) of position s in the portfolio thatresults from offsetting a day-i−1 action action(s, i−1) on a position inasset s is given by

${w^{\prime}\left( {s,i} \right)} = \frac{{w\left( {s,i} \right)} - {{action}\left( {s,{i - 1}} \right)}}{1 - {{action}\left( {s,{i - 1}} \right)}}$

Now, recall our assumption that the adjusted portfolio in which asset shas this weight results from two components. One of those components isthe actual portfolio weighted by the weight value w(o, i) that we areseeking. The other is an offset portfolio weighted by 1−w(o, i). Alsorecall that position s's weight in the actual portfolio is w(s, i) andthat its weight in the offset portfolio it is 100% (in the case of asingle-action offset). We can conclude from these facts that anotherexpression for asset s's weight w′(s, i) in the adjusted portfolio is:w′(s,i)=w(o,i)·w(s,i)+[1−w(o,i)]·1.0.By solving for w(o, i) in the equation that results from equating theprevious two equations' expressions for w′(s, i), we obtain thefollowing value for the weight w(o, i) we are seeking:

${{w\left( {o,i} \right)} = \frac{1}{1 - {{action}\left( {s,{i - 1}} \right)}}},$As we stated above, that value is the actual-portfolio weight in ourequation r(o, i)=w(o, i)·pr(i)+ or(o, i) for the day-i return of theadjusted portfolio that results from an offset o of a single action thatoccurred on day i−1. If offset o offsets only a single-action, in assets, thenor(o,i)=[1−w(o,i)]·r(s,i)

As was mentioned above, the values that we have just determined arethose for the day after the action being offset. Values for subsequentdays are similar, but the weights are adjusted for relative returns:

${w\left( {o,j} \right)} = {{w\left( {o,{j - 1}} \right)}{\frac{1 + {{pr}\left( {j - 1} \right)}}{1 + {r\left( {o,{j - 1}} \right)}}.}}$

We now turn to how quantities determined in the manner described abovecan be combined to arrive at an adjusted portfolio that containscompensation for multiple actions. We will assume, that is, that thereare offsets o1 and o2 for which we have already determined respectiveoffset return contributions or(o1, i) and or(o2, i) and respectiveportfolio weights w(o1, i) and w(o2, i). Although we have so fardemonstrated only how to obtain these quantities for offsets of singleactions, it will become apparent that the following explanation appliesequally to cases in which one or both offsets o1 and o2 providecompensation for multiple-action sets.

What we want is a way to obtain from these values the return r(o3, i) ofthe adjusted portfolio that results from an offset o3 for the union ofo1's and o2's respective action sets. Our approach to doing so will beto postulate that this portfolio's return can be computed, just as asingle-action offset can, as the weighted sum of the actual portfolio'sreturn and some offset return, which in this case we will refer to asor(o3, i). We will postulate, that is, that:r(o3,i)=w(o3,i)·pr(i)+or(o3,i),and we will further postulate that or(o3, i) is some weighted sum ofor(o1, i) and or(o2, i). What remains is to find expressions for w(o3,i) and or(o3, i).

Now, in accordance with our assumption, the actual portfolio return'scontribution to the adjusted portfolio's return will be w(o3, i)·pr(i).We know that or(o1, i) and or(o2, i) are the quantities that, when addedto the quantities w(o1, i)·pr(i) and w(o2, i)·pr(i), respectively,compensate in the above-defined manner for the effects of theirrespective action sets in those quantities. So the values required tocompensate for those action sets' effects in w(o3, i)·pr(i) must beor(o1, i)·[w(o3, i)/w(o1, i)] and or(o2, i)·[w(o3, i)/w(o2, i)],respectively, i.e.:

${r\left( {{o\; 3},i} \right)} = {{{w\left( {{o\; 3},i} \right)} \cdot {{pr}(i)}} + {\frac{w\left( {{o\; 3},i} \right)}{w\left( {{o\; 1},i} \right)} \cdot {{or}\left( {{0\; 1},i} \right)}} + {\frac{w\left( {{o\; 3},i} \right)}{w\left( {{o\; 2},i} \right)} \cdot {{or}\left( {{o\; 2},i} \right)}}}$

Since we have assumed that we already know the constituent offsetreturns and weights or(o1, i), or(o2, i), w(o1, i), and w(o2, i), thatequation can be used to compute the desired adjusted-portfolio returnonce we know what w(o3, i) is. And, from the fact that the weights needto add to 100%, i.e., that

${{{w\left( {{o\; 3},i} \right)} + \frac{w\left( {{o\; 3},i} \right)}{w\left( {{o\; 1},i} \right)} + \frac{w\left( {{o\; 3},i} \right)}{w\left( {{o\; 2},i} \right)}} = 1},$we can conclude that

${w\left( {{o\; 3},i} \right)} = {\frac{{w\left( {{o\; 1},i} \right)} \cdot {w\left( {{o\; 2},i} \right)}}{{w\left( {{o\; 1},i} \right)} + {w\left( {{o\; 2},i} \right)} - {{w\left( {{o\; 1},i} \right)} \cdot {w\left( {{o\; 2},i} \right)}}}.}$

To demonstrate how this works, let us assume that the day-i weights ofthe asset-x, asset-y, and asset z positions in our three-positionportfolio are respectively 40%, 40%, and 20% and result from day-i−1actions of +20%, 0%, and −20%, respectively. If we refer to the offsetsfor the actions in those three positions as oX, oY, and oZ,respectively, then:w(oX,i)=125%; or(oX,i)=−0.25·r(x,i)w(oY,i)=100%; or(oY,i)=0·r(y,i)w(oZ,i)=83%; or(oZ,i)=0.17·r(z,i)

Since there was no action in the asset-y position, an adjusted portfoliowhose weights are all the no-action weights should result from combiningthe individual-position offsets oX and oZ. To show that such an adjustedportfolio does result, we note that the above equation for the combinedoffset oYZ's portfolio weight gives w(oXZ, i)=100%. Using this value andthe other weight values above after substituting oX, oZ, and oXZ for o1,o2, and o3, respectively, in the above equation for the return r(o3, i)of a portfolio adjusted for two constituent offsets yields:r(oXZ,i)=1.00·pr(i)+0.8·or(oX,i)+1.2·or(oZ,i)

Then substituting for the or values yields:r(oXZ,i)=1.00·pr(i)−0.2·r(x,i)+0.2·r(z,i)

And, noting that the actual portfolio's weights are 40%, 40%, and 20%,we can substitute 0.4·r(x, i)+0.4·r(y, i)+0.2·r(z, i) into for theactual-portfolio return pr(i) to obtain:r(oXZ,i)=0.2·r(x,i)+0.4·r(y,i)+0.4·r(y,i),i.e., the return of a portfolio whose positions have the actualportfolio's no-action weights, namely, 20%, 40%, and 40%. In short,combining offsets for all of a day's actions results in anall-no-action-weight portfolio, even though implementing an offset for asingle action does not by itself always return the position in which theaction occurred to its no-action weight.

Now, recall that we described an offset o's offset return contributionor(o, i) as the value such that the return r(o, i) of the portfolio thatresults from adjusting for offset o's actions is given by:r(o,i)=w(o,i)·pr(i)+or(o,i).

From this equation and the above equation for the adjusted portfolio'sreturn r(o3, i), it is apparent that the computation of r(o3, i) canconveniently be computed in such a manner as to produce or(o3, i)

${{or}\left( {{o\; 3},i} \right)} = {{\frac{w\left( {{o\; 3},i} \right)}{w\left( {{o\; 1},i} \right)} \cdot {{or}\left( {{o\; 1},i} \right)}} + {\frac{w\left( {{o\; 3},i} \right)}{w\left( {{o\; 2},i} \right)} \cdot {{or}\left( {{o\; 2},i} \right)}}}$as an intermediate result. By simply retaining this value and that ofweight w(o3, i), it is possible to combine r(o3, i) with anotheroffset's corresponding quantities to find the return of a portfolioadjusted for a superset the actions that o3 offsets. So computing thereturn that results from rolling back n actions is a problem of onlyO(n) complexity.

Now, it is also sometimes valuable to know whether the adjustments beingmade would result in a portfolio whose turnover is greater or less thanthe actual portfolio's. Therefore, some embodiments will also keep trackof the change inc(o, k) in turnover that a given offset causes. In thecase of an offset o for a sell in asset s on day i, that change day i isthat sell's value:inc(o,i)=−sell(s,i).

On subsequent days, it is given byinc(o,i)=scaleUp(sell(s,i),s,i,i+p)where scaleUp(a, s, i, j) is the value of the action that would have tobe taken in asset s's position on day j to give that position's day-jweight the value it would have if an action of value a had occurred onday i:

${{scaleUp}\left( {a,s,i,j} \right)} = {a \cdot {\frac{1 + {r\left( {s,i,j} \right)}}{1 + {\left( {1 - a} \right) \cdot {{pr}\left( {i,j} \right)}} + {a \cdot {r\left( {s,i,j} \right)}}}.}}$

When offsets are combined, the turnover increments simply add:inc(o3,i)=inc(o1,i)+inc(o2,i)

Having now described an advantageous approach to computing adjustedreturns, we return to the operation that FIG. 3's block 62 representsand that FIG. 4 depicts in more detail. As FIG. 4's Block 64 indicates,the illustrated embodiment begins with the value of the first asset onthe first day in the interval for which the computations are to be made,and, as block 66 indicates, it compares the position's age with a targetholding period for the adjusted portfolio as a whole. In order to arriveat an adjusted portfolio in which the average age is the same as theactual portfolio's, this target holding period is set to be twice theportfolio's average age, which in the illustrated embodiment is computedas set forth above. As block 68 indicates, the routine so adjusts thehypothetical portfolio's day-i return as to offset that position ifasset s's age exceeds the target holding period.

Now, rather than an offset of the type described above, which is used tooffset an action and typically affects the hypothetical portfolio'sreturn for more than one day, the type of offset applied here is for aposition and affects only the return of a single day. The discussionabove of how to combine offsets remains valid, but the portfolio-weightvalue is determined from the position weight rather that the action.Consequently, we can represent the resultant determinations of theactual portfolio weight w(o, k) and the offset-portfolio return d(o, k)used in arriving at the adjusted portfolio results for any given day kas follows:

${d\left( {o,k} \right)} = \left\{ {{\begin{matrix}0 & \left. {{for}\mspace{14mu} k}\Leftarrow i \right. \\{r\left( {s,k} \right)} & {{{for}\mspace{14mu} k} = {i + 1}} \\0 & {{{for}\mspace{14mu} k} > {i + 1}}\end{matrix}{w\left( {o,k} \right)}} = {\begin{Bmatrix}1 & \; \\{1 + \frac{w\left( {s,{i + 1}} \right)}{1 - {w\left( {s,{i + 1}} \right)}}} & {{{if}\mspace{14mu}{ages}\mspace{14mu}\left( {s,i} \right)} > {holdingPeriod}} \\1 & {otherwise} \\1 & \;\end{Bmatrix}\begin{matrix}{{{for}\mspace{14mu} k} \leq i} \\{{{for}\mspace{14mu} k} = {i + 1}} \\{{{for}\mspace{14mu} k} > i}\end{matrix}}} \right.$That is, the only day k for which this offset affects the adjustedportfolio's results is day k=i+1. And, in the simple turnover measure weare assuming for the illustrated embodiment, this offset has no effecton that measure:inc(o,k)=0 for all k

For a buy-and-hold position, a similar single-day offset for asset swill be implemented for each subsequent day as the routine loops throughthose days, and the resultant string of single-day offsets has the sameeffect as selling the position on the first day in which its age in theactual portfolio exceeds the target holding period; i.e., the sameeffect could be obtained by an action offset. At least for positions inwhich multiple buys have occurred, though, we consider using multipleposition offsets to be more convenient.

If the result of the block-66 determination is instead that theposition's age does not exceed the target holding period, then theposition should not be offset. As block 70 indicates, though, theroutine in that case determines whether a sell action in that positionoccurred on the day in question. Now, the equation set forth above fordetermining a position's age indicates that a partial sale of a positionthat was acquired over a plurality of days is treated as though assetsfrom each of the buys that contributed to the position are being soldproportionately. To a degree, this convention is arbitrary, and otherembodiments may treat such sales differently for age-determinationpurposes; first-in, first-out or last-in, first-out approaches may betaken, for instance. But the example approach is convenient, because itresults in an age quantity that a sell does not affect, and, as block70's negative branch indicates, no operation to adjust the hypotheticalportfolio has to be taken in response to a sale of a position whose ageexceeds the target holding period.

If the target position's age is less than the target holding period,though, the hypothetical portfolio should not have a sell in thatposition. So, if the actual portfolio does have a sell in that case, theroutine applies an action offset whose effects extend for the number ofdays p by which the intended holding period exceeds the position's ageon the day currently under consideration. That is, block 72 representsmaking adjustments for each day's hypothetical-portfolio return from theday i under consideration to day i+p. The following quantitiescharacterize such an offset:

${d\left( {o,k} \right)} = \left\{ {{\begin{matrix}0 & \left. {{for}\mspace{14mu} k}\Leftarrow i \right. \\{r\left( {s,k} \right)} & {{{{for}\mspace{14mu} k} = {i + 1}},{i + 2},\ldots\mspace{11mu},{i + p}} \\0 & {{{for}\mspace{14mu} k} > {i + p}}\end{matrix}{w\left( {o,k} \right)}} = \left\{ {{\begin{matrix}1 & \left. {{for}\mspace{14mu} k}\Leftarrow i \right. \\{1 - \frac{{sell}\left( {s,i} \right)}{1 + {{sell}\left( {s,i} \right)}}} & {{{for}\mspace{14mu} k} = {i + 1}} \\{{w\left( {o,{k - 1}} \right)} \cdot \frac{1 + {{pr}\left( {k - 1} \right)}}{1 + {r\left( {o,{k - 1}} \right)}}} & {{{{for}\mspace{14mu} k} = {i + 2}},\ldots\mspace{11mu},{i + p}} \\1 & {{{for}\mspace{14mu} k} > {i + p}}\end{matrix}{{inc}\left( {o,k} \right)}} = \left\{ {\begin{matrix}{- {{sell}\left( {s,i} \right)}} & {{{for}\mspace{14mu} k} = i} \\{{scaleUp}\left( {{{sell}\left( {s,i} \right)},s,i,{i + p}} \right)} & {{{for}\mspace{14mu} k} = {i + p}} \\0 & {{for}\mspace{14mu}{all}\mspace{14mu}{other}\mspace{14mu} k}\end{matrix}.} \right.} \right.} \right.$

As blocks 74 and 76 indicate, FIG. 4's item-processing loop for thecurrent day is repeated for every position. And, as blocks 82, 84, and86 indicate, this operation of adding offsets is repeated for each dayi.

Now, the block-68 and block 72 operations of “adding offsets” may insome embodiments include actually computing the hypothetical-portfolioreturns r(o, i) for each offset. But this is not necessary for allpurposes, so those steps may in other embodiments involve computing onlythe weights w(o, i) and offset returns or(o, i). Such embodiments wouldrequire a further operation, represented by block 84, in which eachday's return is computed from the actual return for that day by usingthe offset return and weight of the last-computed offset for that day.

This completes the FIG. 4 routine and thus the operation of FIG. 3'sblock-62 step. With the block-62 operation of computing the unbiasedadjusted portfolio thus completed, the FIG. 3 routine turns toperforming a similar operation for the biased adjusted portfolio, asblock 88 indicates. This operation is the same as the one just describedin connection FIG. 4 for the unbiased adjusted portfolio, with theexception that two different holding periods are applied; one forwinners and another for losers. Specifically, if unrealizedMargin(s, k)exceeds zero (or the tiling threshold), the portfolio's position inasset s on day k is considered a winner, and the holding period used inthe operations that FIG. 4's Blocks 66, 70, and 72 represent equalstwice that of the winner-position average over the interval of interestas whole. If the unrealizedMargin(s, k) value is instead less than zero(or the tiling threshold), then that holding period is instead the onedetermined for the loser positions.

Note here that a given position can be both a winner and a loser atdifferent times throughout the interval of interest and that theequation given above for average age reflects this. Specifically, thereis not just a single contribution to the average-age quantity for eachportfolio position; instead, there is a separate contribution for eachcombination of position and day. So contributions for a given positioncan be made to the winner group's average age for some days and to theloser group's average age for other days.

With the different portfolios' returns thus determined, the systemgenerates an output, as block 90 indicates, that represents the returncomparison. That output may merely set forth the different portfolios'returns, or it could produce an output that is determined by them, suchas their difference, ratio, etc.

Outputs that compare the returns on a risk-adjusted basis areparticularly helpful. Numerous ways of adjusting returns for risk areknown to those skilled in the art, as is evidenced by papers such as“Understanding Risk and Return, the CAPM and the Fama-FrenchThree-Factor Model” by Adam Borchert, Lisa Ensz, Joep Knijn, Greg Popeand Aaron Smith, Tuck School of Business at Dartmouth (DartmouthCollege), Case Note 03-111 (2003), which describes the CAPM and thethree-factor model. Various embodiments may use such approaches, onethat results from adding a momentum factor to the three-factor modeland/or others to generate outputs based on the impact calculations.

Now, the hypothetical adjusted portfolio that results from treating agiven asset as a winner on some days and a loser on others can in somecases be one in which that asset is bought and sold more frequently thanis realistic. If the holding period for winners is 400 days and theperiod for losers is 500 days, for instance, then a given position whoseunrealized margin waivers around zero during the interval in which itsage increases from 400 day to 500 days will be represented in thehypothetical portfolio as being bought and sold many times during that100 days. Although this is not realistic behavior for an actualportfolio, it actually serves the adjusted portfolio's purpose well. Ityields a result that is a weighted average of winner and loser resultsin proportion to the degree to which the position exhibited winner andloser behavior.

Note that the hypothetical portfolio's buy-action dates are the same asthose of the actual portfolio in the above approach; it is the effectivesell-action dates that change. Although we prefer this approach, otherembodiments of the invention may not follow that rule. For example,equal holding periods can be achieved by instead advancing buy actionsand keeping the sell-action dates the same or by various combinations ofadvancing or delaying buy actions and advancing or delaying sell actionsin combinations that may be different for different classes.

The offset quantities that may be used to advance a sale rather thandelay it are given by:

${d\left( {o,k} \right)} = \left\{ {{\begin{matrix}0 & \left. {{for}\mspace{14mu} k}\Leftarrow{i - p} \right. \\{r\left( {s,k} \right)} & {{{{for}\mspace{14mu} k} = {i - p + 1}},{i - p + 2},\ldots\mspace{11mu},i} \\0 & {{{for}\mspace{14mu} k} > i}\end{matrix}{w\left( {o,k} \right)}} = \left\{ {{\begin{matrix}1 & \left. {{for}\mspace{14mu} f}\Leftarrow{i - p} \right. \\{1 + \frac{{scaleBack}\left( {{{sell}\left( {s,i} \right)},s,{i - p},i} \right)}{1 - {{scaleBack}\left( {{{sell}\left( {s,i} \right)},s,{i - p},i} \right)}}} & {{{for}\mspace{14mu} k} = {i - p + 1}} \\{{w\left( {o,{k - 1}} \right)} \cdot \frac{1 + {{pr}\left( {k - 1} \right)}}{1 + {r\left( {o,{k - 1}} \right)}}} & {{{{for}\mspace{14mu} k} = {i - p + 2}},\ldots\mspace{11mu},i} \\1 & {{{for}\mspace{14mu} k} > i}\end{matrix}{{inc}\left( {o,k} \right)}} = \left\{ \begin{matrix}{{scaleBack}\left( {{{sell}\left( {s,i} \right)},s,{i - p},i} \right)} & {{{for}\mspace{14mu} k} = {i - p}} \\{- {{sell}\left( {s,i} \right)}} & {{{for}\mspace{14mu} k} = i} \\0 & {{for}\mspace{14mu}{all}\mspace{14mu}{other}\mspace{14mu} k}\end{matrix} \right.} \right.} \right.$where scaleBack(a, s, i, j) is the value of the action that would havehad to be taken in asset s's position on day i to give that position'sweight on day j the value that an action of value a on day j wouldcause:

${{scaleBack}\left( {a,s,i,j} \right)} = {a \cdot {\frac{1 + {{pr}\left( {i,j} \right)}}{1 + {r\left( {s,i,j} \right)}}.}}$

Examples of quantities that can be used for delaying a buy are:

${d\left( {o,k} \right)} = \left\{ {{\begin{matrix}0 & \left. {{for}\mspace{14mu} k}\Leftarrow i \right. \\{r\left( {s,k} \right)} & {{{{for}\mspace{14mu} k} = {i + 1}},{i + 2},\ldots\mspace{11mu},{i + p}} \\0 & {{{for}\mspace{14mu} k} > {i + p}}\end{matrix}w\left( {o,k} \right)} = \left\{ {{\begin{matrix}1 & \left. {{for}\mspace{14mu} k}\Leftarrow i \right. \\{1 + \frac{{buy}\left( {s,i} \right)}{1 - {{buy}\left( {s,i} \right)}}} & {{{for}\mspace{14mu} k} = {i + 1}} \\{{w\left( {o,{k - 1}} \right)} \cdot \frac{1 + {{pr}\left( {k - 1} \right)}}{1 + {r\left( {o,{k - 1}} \right)}}} & {{{{for}\mspace{14mu} k} = {i - p + 2}},\ldots\mspace{11mu},i} \\1 & {{{for}\mspace{14mu} k} > i}\end{matrix}{{inc}\left( {o,k} \right)}} = {0\mspace{14mu}{for}\mspace{14mu}{all}\mspace{14mu} k}} \right.} \right.$

And the quantities for advancing a buy are:

${d\left( {o,k} \right)} = \left\{ {{\begin{matrix}{0} & {{{for}\mspace{14mu} k}<={i - p}} \\{r\left( {s,k} \right)} & {{{{for}\mspace{14mu} k} = {i - p + 1}},{i - p + 2},\ldots\mspace{11mu},i} \\{0} & {{{for}\mspace{14mu} k} > i}\end{matrix}{w\left( {o,k} \right)}} = \left\{ {{\begin{matrix}{1} & {{{for}\mspace{14mu} k}<={i - p}} \\{1 - \frac{{buy}\left( {s,i} \right)}{1 + {{buy}\left( {s,i} \right)}}} & {{{for}\mspace{14mu} k} = {i - p + 1}} \\{{w\left( {o,{k - 1}} \right)} \cdot \frac{1 + {{pr}\left( {k - 1} \right)}}{1 + {r\left( {o,{k - 1}} \right)}}} & {{{{for}\mspace{14mu} k} = {i - p + 2}},\ldots\mspace{11mu},i} \\1 & {{{for}\mspace{14mu} k} > i}\end{matrix}{inc}\left( {o,k} \right)} = {0\mspace{14mu}{for}\mspace{14mu}{all}\mspace{14mu} k}} \right.} \right.$

Note that these example delaying and advancing equations treat sells andbuys differently. The buy treatment is based on the assumption that themanager wants to add some predetermined additional weight in thatposition, and this weight increment is preserved when the buy isadvanced or delayed: the same weight is bought p days earlier. Incontrast, the sell treatment is based on the assumption that thedecision is to sell some proportion of the position itself, and this iswhat is preserved.

Of course, some embodiments' advancing and delaying approaches may bebased on different assumptions. Even with these assumptions, moreover,the above equations result in something of an approximation. Since buysare not scaled when they are advanced, for example, there is a residualgain or loss for which the equations above do not account.

This can be appreciated by considering the case of advancing a buy of 50basis points (“bps”) by one month. Suppose that the manager holds thisposition for a year. Advancing that buy will likely result in a positionthat is not exactly 50 bps at the time of the original purchase.Suppose, in fact, that the position grew to 60 bps by the time of theoriginal purchase. To model this effect, the additional 10 bps (60bps-50 bps) needs to be carried forward until the original sell date.And, at the original sell date, the additional carried amount of 10 bps,which may have increased or decreased with respect to the portfolio,needs to be sold. Selling this carry-over will result in additionalturnover for the adjusted portfolio. There is also a (much smaller)carry-over for sells. So some embodiments may include provisions forreflecting them, but our experience is that any increased accuracy istoo small to justify the added computation.

As was mentioned above, another way to modulate actions is to changetheir relative sizes. The following quantities can be used to implementoffsets for this purpose:

${d\left( {o,k} \right)} = \left\{ {{\begin{matrix}{0} & {{{for}\mspace{14mu} k}<=i} \\{r\left( {s,k} \right)} & {{{{for}\mspace{14mu} k} = {i + 1}},{i + 2},\ldots\mspace{11mu},{i + {{life}\left( {s,i} \right)}}} \\{0} & {{{for}\mspace{14mu} k} > {i + {{life}\left( {s,i} \right)}}}\end{matrix}{w\left( {o,k} \right)}} = \left\{ {{\begin{matrix}{1} & {{{for}\mspace{14mu} k}<=i} \\{1 - \frac{{{buy}\left( {s,i} \right)} \cdot \left( {{factor} - 1} \right)}{1 - {{{buy}\left( {s,i} \right)} \cdot \left( {{factor} - 1} \right)}}} & {{{for}\mspace{14mu} k} = {i + 1}} \\{{w\left( {o,{k - 1}} \right)} \cdot \frac{1 + {{pr}\left( {k - 1} \right)}}{1 + {r\left( {o,{k - 1}} \right)}}} & {{{{for}\mspace{14mu} k} = {i + 2}},\ldots\mspace{11mu},{i + {{life}\left( {s,i} \right)}}} \\1 & {{{for}\mspace{14mu} k} > {i + {{life}\left( {s,i} \right)}}}\end{matrix}{{inc}\left( {o,k} \right)}} = \left\{ \begin{matrix}{{scaleUp}\left( {{{{buy}\left( {s,i} \right)} \cdot \left( {{factor} - 1} \right)},s,i,{i + {{life}\left( {s,i} \right)}}} \right)} & {{{for}\mspace{14mu} k} = {i + {{life}\left( {s,i} \right)}}} \\{0} & {{for}\mspace{14mu}{all}\mspace{14mu}{other}\mspace{14mu} k}\end{matrix} \right.} \right.} \right.$

where factor is the ratio of the hypothetical portfolio's action to thecorresponding actual-portfolio action.

Outputs

It is anticipated that most systems that implement the presentinvention's teachings will afford users the capability of choosing amongmany different outputs representing many different quantities. Suppose,for example, that the user is interested in analyzing a portfolio forthe gain effect. The system may give such a user the option not only ofchoosing that effect but also of specifying whether that effect is to beexplored in sells or buys and whether the basis for comparison will betiles or will instead be “splits” based on a predetermined threshold.

And one type of resultant output that the system could present may be alist of statistics such as the one that FIG. 5 depicts. As the “weight”row indicates, the system has used equal-weight tiles as the basis forcomparison and has computed respective values of turnover, age, margin,and contribution for each tile. If the system had instead based itsclassification on predetermined-threshold splits, those weights wouldtypically differ. The FIG. 5 output also displays not only a resultantbias-measure value but also its T-statistic and, as example measures ofbias impact, the differences between the two tiles' returns and alphas.

The system may additionally or instead present results in graphicalform. FIG. 6 depicts an example. There the system has treated two tilesof one actual portfolio as respective individual active portfolios. Foreach tile it has determined performance-measure values for a pluralityof hypothetical portfolios. Each hypothetical portfolio is associatedwith a different target holding period, and its positions are obtainedby so modulating that tile's positions that those positions all have theassociated holding period. The upper-left graph plots those performancemeasures against holding period for the portfolios obtained bymodulating one tile's positions, and the upper-right graph plots thosemeasures for hypothetical portfolios similarly obtained from the othertile's. The example performance measures are the differences in return,alpha, and turnover between those hypothetical portfolios and either theactual portfolio itself or another hypothetical portfolio, whose holdingperiods are uniform but have an average that equals that of the actualportfolio.

FIG. 6's lower graphs are similarly associated with respective tiles,but the hypothetical portfolios there have resulted from time-shiftingactions of a selected type.

Assessing Advantage

Similar plots can be used to provide an indication of what we refer toas the portfolio manager's buying or selling “advantage.” The portfoliomanager typically holds different positions for different lengths oftime, presumably to add value. These advantage values are a way ofmeasuring the degree to which the portfolio manager has achieved thisend. The illustrated embodiment calculates two types of buying andselling advantage: the return-based type and the alpha-based type. Someobservation interval is defined, and the actual portfolio's averageholding period over that interval is determined. The return- andalpha-type buying averages are respectively the return and alpha of ahypothetical portfolio whose positions have a uniform holding periodequal to that average. The return- and alpha-type selling averages arerespectively the differences between the return- and alpha-type buyingadvantages and the actual portfolio's return and alpha.

Such values may be presented by themselves or with other values. Forexample, the system may compare the actual portfolio's returns, alphas,and turnovers with those of a variety of hypothetical portfolios, eachof which is associated with a different holding period, for which all ofthat hypothetical portfolio's positions are held, and it may displaythose values to the user in a display of a type exemplified by FIG. 7.In that drawing, the differences in return, alpha, and turnover betweenthe actual portfolio and hypothetical portfolios having the samepositions but different respective uniform holding periods are plottedagainst the holding period associated with the hypothetical portfolio.

Accounting for Corporate Actions

The concepts of return and realization were left undefined above. Thisis appropriate, because their precise definitions are not critical tothe present invention's teachings, and there will likely be somevariation in which various embodiments treat them. For the sake ofexample, though, we will set forth one approach to treating return'svarious components.

In this approach, the equations for no-action weight, fractionunrealized, and unrealized contribution are modified for spin-offs andspecial dividends, as will be explained below, in such a manner that achange in position weight that results from such a corporate actionshould not be interpreted as a sale but should be considered arealization. To give the modified equations, we introduce the followingquantities, which represent market data typically obtained separatelyfrom the portfolio data:

-   -   price(s, i) is asset s's price at close of day i.    -   incomeAmount(s,i) is asset s's ordinary dividend with ex-day of        i.    -   eventAmount(s,i) is asset s's special dividend or proceeds from        spin-off on ex-day i.    -   splitFactor(s,i) is asset s's split factor on ex-day i.

The equations below allocate stock return and contribution into threeparts: for price changes, for ordinary dividends, and for corporateactions (such as spin-offs and special dividends).

Asset s's return rfp(s, i) from (split-adjusted) price from the close onday i−1 to the close on day i is given by:

${{rfp}\left( {s,i} \right)} = \frac{{{{price}\left( {s,i} \right)} \cdot {{splitFactor}\left( {s,i} \right)}} - {{price}\left( {s,{i - 1}} \right)}}{{price}\left( {i - 1} \right)}$

Asset s's return rfi(s, i) from income from the close on day i−1 to theclose on day i is given by:

${{rfi}\left( {s,i} \right)} = \frac{{incomeAmount}\left( {s,i} \right)}{{price}\left( {i - 1} \right)}$

Asset s's return rfe(s, i) from spin-offs, special dividends, etc. fromthe close on day i−1 to the close on day i is given by:

${{rfe}\left( {s,i} \right)} = \frac{{eventAmount}\left( {s,i} \right)}{{price}\left( {i - 1} \right)}$

Asset s's total return r (s, i) from spin-offs, special dividends, etc.from the close on day i−1 to the close on day i is given by:r(s,i)=rfp(s,i)+rfi(s,i)+rfe(s,i).

Asset s's contribution cfp(s, i) from changes in price on day i is givenby:cfp(s,i)=w(s,i)·rfp(s,i)

Asset s's contribution cfi(s, i) from ordinary dividends on day i isgiven by:cfi(s,i)=w(s,i)·rfi(s,i)

Asset s's contribution cfe(s, i) from special dividends and spin-offs onday i is given by:cfe(s,i)=w(s,i)·rfe(s,i)

Asset s's total contribution c(s, i) on day i is given by:c(s,i)=cfp(s,i)+cfi(s,i)+cfe(s,i)

Asset s's contribution cfpe(s, i) on day i excluding ordinary dividendsis given by:cfpe(s,i)=cfp(s,i)+cfe(s,i)

Below are no-action-weight, fraction-unrealized, andunrealized-contribution equations that reflect this treatment ofspin-offs and special dividends:

$\quad\begin{matrix}{{{naw}\left( {s,i} \right)} = {\frac{1 + {{rfp}\left( {s,{i - 1}} \right)}}{1 + {{pr}\left( {i - 1} \right)}} \cdot {w\left( {s,{i - 1}} \right)}}} \\{{{fu}\left( {s,i} \right)} = {\left( {1 - {{fr}\left( {s,i} \right)}} \right) \cdot \left( {1 - {{rfe}\left( {s,i} \right)}} \right)}} \\{{{uc}\left( {s,i,j} \right)} = \left\lbrack {{u\left( {s,j} \right)} \cdot \left\lbrack {{{uc}\left( {s,i,{j - 1}} \right)} +} \right.} \right.} \\\left. {{fcfpe}{\left( {s,j} \right) \cdot \left( {1 + {{pr}\left( {0,{j - 1}} \right)}} \right)}} \right\rbrack\end{matrix}\mspace{50mu}$Handling Short Positions and Leverage

To this point, portfolios with only long positions have been considered.Handling short positions is a straightforward generalization of theabove concepts. A short position's weight in the actual portfolio isnegative. The actions in short positions will be referred to as shortsand covers rather than buys and sells. Conceptually, a short iscomparable to a buy, and a cover is comparable to a sell; buys andshorts both increase the exposure to a stock, while sells and coversdecrease the exposure. The term increase will be used to denote a buy ora short, and decrease will be used to denote a sell or a cover.

Below are the revised definitions that can be applied to both long andshort positions:

${{sell}\left( {s,i} \right)} = \left\{ {{\begin{matrix}{\min\left( {{{- \Delta}\;{w\left( {s,i} \right)}},{{naw}\left( {s,{i + 1}} \right)}} \right)} & {{{if}\mspace{14mu}\Delta\;{w\left( {s,i} \right)}} < {0\mspace{14mu}{and}\mspace{14mu}{w\left( {s,i} \right)}} > 0} \\{0} & {otherwise}\end{matrix}{{buy}\left( {s,i} \right)}} = \left\{ {{\begin{matrix}{\min\left( {{\Delta\;{w\left( {s,i} \right)}},{w\left( {s,{i + 1}} \right)}} \right)} & {{{if}\mspace{14mu}\Delta\;{w\left( {s,i} \right)}} > {0\mspace{14mu}{and}\mspace{14mu}{w\left( {s,{i + 1}} \right)}} > 0} \\{0} & {otherwise}\end{matrix}{{short}\left( {s,i} \right)}} = \left\{ {{\begin{matrix}{\min\left( {{{- \Delta}\;{w\left( {s,i} \right)}},{- {w\left( {s,{i + 1}} \right)}}} \right)} & {{{if}\mspace{14mu}\Delta\;{w\left( {s,i} \right)}} < {0\mspace{14mu}{and}\mspace{14mu}{w\left( {s,{i + 1}} \right)}} < 0} \\{0} & {otherwise}\end{matrix}{{cover}\left( {s,i} \right)}} = \left\{ {{\begin{matrix}{\min\left( {{{- \Delta}\;{w\left( {s,i} \right)}},{- {{naw}\left( {s,{i + 1}} \right)}}} \right)} & {{{if}\mspace{14mu}\Delta\;{w\left( {s,i} \right)}} > {0\mspace{14mu}{and}\mspace{14mu}{w\left( {s,i} \right)}} < 0} \\{0} & {otherwise}\end{matrix}{{increase}\left( {s,i} \right)}} = {{{buy}\left( {s,i} \right)} - {{short}\left( {s,i} \right)}}} \right.} \right.} \right.} \right.$

The quantity increase(s, i) is positive for long positions and negativefor short positions.decrease(s,i)=sell(s,i)−cover(s,i)

The quantity decrease(s, i) is positive for long positions and negativefor short positions.

The fraction fr(s, i) realized due to an action is then:

${{fr}\left( {s,i} \right)} = \frac{{decrease}\left( {s,i} \right)}{{naw}\left( {s,{i + 1}} \right)}$

And the fraction fp(s, i) purchased due to an action is:

${{fp}\left( {s,i} \right)} = {\frac{{increase}\left( {s,i} \right)}{w\left( {s,{i + 1}} \right)}.}$

The method described above for constructing an adjusted portfolio stillapplies when the actual portfolio has shorts and margin; although theweights of some positions in the actual portfolio become negative, thesum of the weights remains 1.0. But the calculation of the weight

${w\left( {{o\; 3},i} \right)} = \frac{{w\left( {{o\; 1},i} \right)} \cdot {w\left( {{o\; 2},i} \right)}}{{w\left( {{o\; 1},i} \right)} + {w\left( {{o\; 2},i} \right)} - {{w\left( {{o\; 1},i} \right)} \cdot {w\left( {{o\; 2},i} \right)}}}$to be applied to the actual portfolio when effects of two offsets o1 ando2 are combined can be vulnerable to large floating-point errors whensome of the actual portfolio's positions have negative weights; thatdenominator can be very small or zero.

So the calculations are best performed by partitioning the actualportfolio into two sub-portfolios, one containing the long positions andthe other containing the short positions. The offsets are performedwithin the sub-portfolios. This avoids the singularity, eachsub-portfolio has no short positions or leverage. The overalladjusted-portfolio return is obtained by adding together thesub-portfolios' weighted returns.

Turnover Calculation Based on All Actions

It was mentioned above that some embodiments may keep track of how muchof a difference in turnover would result from the adjusted portfolio,and an approach that evaluated turnover in terms of sells only wasdescribed. Turnover calculations can alternatively be made in a way thatreflects all actions, including shorts and covers.

Here are quantities related to turnover evaluated in this way:

${{decreases}(i)} = {{\sum\limits_{s \in S}{{sell}\left( {s,i} \right)}} + {{cover}\left( {s,i} \right)}}$${{{in}{creases}}(i)} = {{\sum\limits_{s \in S}{{buy}\left( {s,i} \right)}} + {{cover}\left( {s,i} \right)}}$

A turnover value based on decreases is given by:

${{decreaseTurns}\left( {i,j} \right)} = {\sum\limits_{k = i}^{j}{{decreases}(k)}}$and an annualized version is given by:

${{decreaseTurnovers}\left( {i,j} \right)} = {\frac{\sum\limits_{k = i}^{j}{{decreases}(k)}}{j - i + 1} \cdot 252}$

A turnover value based on increases is given by:

${{increaseTurns}\left( {i,j} \right)} = {\sum\limits_{k = i}^{j}{{increases}(k)}}$and an annualized version is given by:

${{increaseTurnovers}\left( {i,j} \right)} = {\frac{\sum\limits_{k = i}^{j}{{increases}(k)}}{j - i + 1} \cdot 252}$

To calculate increaseTurns and decreaseTurns for an adjusted portfolio,two inc time-series have to be maintained and populated: decreaseIncsand increaseIncs. These values' computations are similar, mutatismutandis, to those of the inc values described above.

As was mentioned above, a computer system that implements the presentinvention's teachings may afford the user the option of selecting fromamong many of the quantities defined above, or quantities derived fromthem, as outputs that the system is to provide as measures of aportfolio's performance. This is true not only of the actual portfoliothat the input data describe but also of hypothetical portfolios derivedfrom them. Similarly, although the foregoing description hasconcentrated on hypothetical portfolios determined in specific ways forspecific purposes, some embodiments may afford the user the option ofcustomizing the hypothetical portfolios in a wide variety of ways.

Some, for example, may enable the user to choose at a particularly finegranularity the changes that will be used to produce a hypotheticalportfolio from an actual one. The user may be able to select anindividual action, e.g., the action that occurred in a stated stock on astated day, and specify a specific type of modulation, e.g., a delay ofthat action by five days.

Some embodiments may enable the user to specify the changes at a higherlevel. The user may, for instance, be given the option of selecting amodulation type and of selecting actions by, say, action type, timeinterval, and position attribute. For example, the user may elect toshifting all buys (or sells, shorts, covers, increases, or decreases)that occurred during the year 2004 in non-cash positions whose ages areless than, say, 63 market days. Instead of or in addition to age, theuser may be able to select from among any of the other attributesmentioned above, as well as others, or any combination of them. In eachcase, the user could specify upper and/or lower limits on thoseattributes in terms of fixed values or in terms of, say, the relativesize of the action or position group that would thereby qualify formodulation.

For some attributes, the user may also be afforded the option ofspecifying additional parameters. If the attribute is unrealized margin,for example, the user may specify an “anchor” value to indicate theperiod over which the unrealized margin is to be evaluated. The anchorvalue would be used as the threshold or quasi-basis for determining gainor loss; an anchor value of −20, for example, might mean that the userwants the basis for qualification to be short-term unrealized marginover the last twenty days.

The user will typically be afforded the option of selecting not only theactions and/or positions to be modulated but also the type ofmodulation, e.g., of specifying whether the modulation is to be a shiftor a change in size. As was the case with the set-holding-period examplegiven above, the system may meet some user requests by employing acombination of modulation types, such as (in that case, delay-type)shifts and position offsets.

If the type of modulation is a shift, the user may specify a fixedshift; as was mentioned above; for example, the user may requestperformance measures for each of a plurality of hypothetical portfolios,each of which results from a different fixed shift. Or, as wasexemplified above, the user may specify that the shifts so vary amongpositions that as many of those positions' holding periods as possibleequal the same set value. The user may also be given the capability ofselecting whether the shifts to be used for this purpose advance ordelay actions.

If the type of modulation is a change in size, the system may afford theuser the option of specifying a fixed value or ratio for the sizechange. By specifying a “factor” input, for example, the user mayspecify that the hypothetical portfolio should result from actions whosevalues are products of that factor's value and correspondingactual-portfolio actions. (The factor's value would typically but notnecessarily be constrained to be positive.)

The actual portfolio on which a given hypothetical portfolio is basedmay be one that consists of a user-selected subset of some larger actualportfolio's positions. And, just as the user can choose actions formodulation in accordance with the attribute values exhibited by thepositions in which those actions occur, the system may afford the userthe capability of defining these subsets in accordance with theirpositions' attribute values. For this purpose, the system may enable theuser to select the number of smaller actual portfolios into which topartition the larger actual portfolio. And it may offer the user theoption of partitioning the larger portfolio's positions in accordancewith fixed thresholds in the chosen attribute's value or to have thesystem itself so choose the thresholds as to result in “tiles,” whoseweights in the larger actual portfolio are equal or have some otherpredetermined ratio. The “weight” used for this purpose may be based onthe positions' weights on some common day or on some other quantity,such as their respective cost bases.

As was mentioned above, the user will typically be given the option ofspecifying the performance measures that the system should compute forthe chosen hypothetical portfolios. Example performance measures arereturn, risk (e.g., standard deviation of monthly returns),risk-adjusted return (as calculated in accordance with a techniqueselected by the user from among, say one-, three-, and four-factormodels), turnover, and information ratio (the ratio of average monthlyrelative return to the standard deviation of monthly relative returns).Outputs based on such performance measures may be the values of themeasures themselves or comparisons of those values with thecorresponding values determined for other portfolios or market indexes.Among the type of comparison may be simple differences, ratios,side-by-side displays, or other ways of describing one quantity byreference to another.

The present invention enables a portfolio's performance to be assessedin ways that are particularly sensitive to tactical aspects. It thusconstitutes a significant advance in the art.

1. A portfolio-analysis system comprising: at least one computerconfigured to: A) receive inputs representing portfolio data thatspecify, for each of a sequence of record times, an actual portfolio'spositions in respective assets; and B) respond to a request for anindication of advantage by: i) identifying relevant actions onrespective said positions in the portfolio data; ii) determining for aperformance measure the values thereof exhibited by the actual portfolioand at least one hypothetical portfolio whose positions result from somodulating the actions of the actual portfolio that the averagemagnitude of the differences between the holding periods of thehypothetical portfolio's positions and those holding periods' average isless than that of the actual portfolio; and iii) generating an outputbased on a comparison of the values thereby determined for the actualportfolio and at least one said hypothetical portfolio.
 2. Aportfolio-analysis system as defined in claim 1 wherein the performancemeasure is portfolio return.
 3. A portfolio-analysis system as definedin claim 1 wherein the relevant actions are identified by detectingdifferences meeting a set of at least one criterion between: A) theweights of positions in the actual portfolio on those record times; andB) those positions' weights on previous record times adjusted for thosepositions' returns.
 4. A portfolio-analysis system as defined in claim 1wherein: A) the value of the performance measure is determined for thehypothetical portfolio over an observation period; and B) at least oneof the non-cash assets in which the hypothetical portfolio holdspositions at some time during the observation period is among thenon-cash assets in which the actual portfolio holds positions at one ormore times during some period included in the sequence of record times.5. A portfolio-analysis system as defined in claim 4 wherein the atleast one of the non-cash assets in which the hypothetical portfolioholds positions at one or more times during the observation period isamong the non-cash assets in which the actual portfolio holds positionsat one or more times during the observation period.
 6. Aportfolio-analysis system as defined in claim 1 wherein the output isgenerated from the hypothetical portfolio's value of the performancemeasure by making a comparison of that value with the actual portfolio'svalue of that performance measure.
 7. A portfolio-analysis system asdefined in claim 1 wherein the output is generated from the hypotheticalportfolio's value of the performance measure by making a comparison ofthat value with a value of that performance measure determined for asecond hypothetical portfolio that comprises assets drawn from those ofthe actual portfolio by modulating the actions of the actual portfolio.8. A portfolio-analysis system as defined in claim 7 wherein thefirst-mentioned hypothetical portfolio's positions result from somodulating the actual portfolio's actions that the average magnitude ofthe differences between the holding periods of the hypotheticalportfolio's positions and their holding periods' average is less thanthat of the actual portfolio.
 9. A portfolio-analysis system as definedin claim 8 wherein the first-mentioned hypothetical portfolio'spositions result from establishing a common target holding period,delaying sales of positions whose ages are less than a target holdingperiod, and eliminating positions whose ages exceed the target holdingperiod.
 10. A portfolio-analysis system as defined in claim 1 whereinevery non-cash asset in which the hypothetical portfolio holds aposition at one or more record times is among the assets in which theactual portfolio holds positions at one or more record times.
 11. Aportfolio-analysis system as defined in claim 10 wherein thehypothetical portfolio's positions result from so modulating the actualportfolio's actions that the average magnitude of the differencesbetween the holding periods of the hypothetical portfolio's positionsand their holding periods' average is less than that of the actualportfolio.
 12. A portfolio-analysis system as defined in claim 10wherein the hypothetical portfolio's positions result from so modulatingthe actual portfolio's actions that the hypothetical portfolio exhibitsactions at a plurality of the record times that differ from those thatthe actual portfolio exhibits.
 13. A portfolio-analysis system asdefined in claim 12 wherein at least one said position attribute inaccordance with which each said action's respective position isclassified is that position's unrealized margin up to the record time ofthat action.
 14. A computer-readable storage medium containing machineinstructions that when executed by a computer system that includes atleast one computer causes the computer system to: A) receive inputsrepresenting portfolio data that specify, for each of a sequence ofrecord times, an actual portfolio's positions in respective assets; andB) respond to a request for an indication of advantage by: i)identifying relevant actions on respective said positions in theportfolio data; ii) determining for a performance measure the valuesthereof exhibited by the actual portfolio and at least one hypotheticalportfolio whose positions result from so modulating the actions of theactual portfolio that the average magnitude of the differences betweenthe holding periods of the hypothetical portfolio's positions and thoseholding periods' average is less than that of the actual portfolio; andiii) generating an output based on a comparison of the values therebydetermined for the actual portfolio and at least one said hypotheticalportfolio.
 15. A storage medium as defined in claim 14 wherein theperformance measure is portfolio return.
 16. A storage medium as definedin claim 14 wherein the relevant actions are identified by detectingdifferences meeting a set of at least one criterion between: A) theweights of positions in the actual portfolio on those record times; andB) those positions' weights on previous record times adjusted for thosepositions' returns.
 17. A storage medium as defined in claim 14 wherein:A) the value of the performance measure is determined for thehypothetical portfolio over an observation period; and B) at least oneof the non-cash assets in which the hypothetical portfolio holdspositions at some time during the observation period is among thenon-cash assets in which the actual portfolio holds positions at one ormore times during some period included in the sequence of record times.18. A storage medium as defined in claim 17 wherein at least one of thenon-cash assets in which the hypothetical portfolio holds positions atone or more times during the observation period is among the setnon-cash assets in which the actual portfolio holds positions at one ormore times during the observation period.
 19. A storage medium asdefined in claim 14 wherein the output is generated from thehypothetical portfolio's value of the performance measure by making acomparison of that value with the actual portfolio's value of thatperformance measure.
 20. A storage medium as defined in claim 14 whereinthe output is generated from the hypothetical portfolio's value of theperformance measure by making a comparison of that value with a value ofthat performance measure determined for a second hypothetical portfoliothat comprises assets drawn from those of the actual portfolio bymodulating the actions of the actual portfolio.
 21. A storage medium asdefined in claim 20 wherein the first-mentioned hypothetical portfolio'spositions result from so modulating the actual portfolio's actions thatthe average magnitude of the differences between the holding periods ofthat hypothetical portfolio's positions and their holding periods'average is less than that of the actual portfolio.
 22. A storage mediumas defined in claim 21 wherein the first-mentioned hypotheticalportfolio's positions result from establishing a common target holdingperiod, delaying sales of positions whose ages are less than a targetholding period, and eliminating positions whose ages exceed the targetholding period.
 23. A storage medium as defined in claim 14 whereinevery non-cash asset in which the hypothetical portfolio holds aposition at one or more record times is among the assets in which theactual portfolio holds positions at one or more record times.
 24. Astorage medium as defined in claim 23 wherein the hypotheticalportfolio's positions result from so modulating the actual portfolio'sactions that the average magnitude of the differences between theholding periods of the hypothetical portfolio's positions and theirholding periods' average is less than that of the actual portfolio. 25.A storage medium as defined in claim 23 wherein the hypotheticalportfolio's positions result from so modulating the actual portfolio'sactions that the hypothetical portfolio exhibits actions at a pluralityof the record times that differ from those that the actual portfolioexhibits.
 26. A storage medium as defined in claim 25 wherein at leastone said position attribute in accordance with which each said action'srespective position is classified is that position's unrealized marginup to the record time of that action.
 27. A method of providing aportfolio-performance assessment comprising: A) employing a computersystem that includes at least one computer to receive inputsrepresenting portfolio data that specify, for each of a sequence ofrecord times, an actual portfolio's positions in respective assets; andB) employing the computer system to respond to a request for anindication of advantage by: i) identifying relevant actions onrespective said positions in the portfolio data; ii) determining for aperformance measure the values thereof exhibited by the actual portfolioand at least one hypothetical portfolio whose positions result from somodulating the actions of the actual portfolio that the averagemagnitude the differences between the holding periods of thehypothetical portfolio's positions and those holding periods' average isless than that of the actual portfolio; and iii) generating an outputbased on a comparison of the values thereby determined for the actualportfolio and at least one said hypothetical portfolio.
 28. A method asdefined in claim 27 wherein every non-cash asset in which thehypothetical portfolio holds a position at one or more record times isamong the assets in which the actual portfolio holds positions at one ormore record times.
 29. A method as defined in claim 28 wherein thehypothetical portfolio's positions result from so modulating the actualportfolio's actions that the average magnitude of the differencesbetween the holding periods of the hypothetical portfolio's positionsand their holding periods' average is less than that of the actualportfolio.
 30. A method as defined in claim 28 wherein the hypotheticalportfolio's positions result from so modulating the actual portfolio'sactions that the hypothetical portfolio exhibits actions at a pluralityof the record times that differ from those that the actual portfolioexhibits.
 31. A method as defined in claim 30 wherein at least one saidposition attribute in accordance with which each said action'srespective position is classified is that position's unrealized marginup to the record time of that action.